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George Frideric Handel

George Frideric (or Frederick) Handel (baptised Georg Fried(e)rich Händel, ; 23 February 1685 – 14 April 1759) was a German-British Baroque composer well known for his operas, oratorios, anthems, concerti grossi, and organ concertos. Handel received his training in Halle and worked as a composer in Hamburg and Italy before settling in London in 1712, where he spent the bulk of his career and became a naturalised British subject in 1727. He was strongly influenced both by the middle-German polyphonic choral tradition and by composers of the Italian Baroque. In turn, Handel's music forms one of the peaks of the "high baroque" style, bringing Italian opera to its highest development, creating the genres of English oratorio and organ concerto, and introducing a new style into English church music. He is consistently recognized as one of the greatest composers of his age.

Handel started three commercial opera companies to supply the English nobility with Italian opera. In 1737, he had a physical breakdown, changed direction creatively, addressed the middle class and made a transition to English choral works. After his success with Messiah (1742), he never composed an Italian opera again. His orchestral Water Music and Music for the Royal Fireworks remain steadfastly popular. One of his four coronation anthems, Zadok the Priest, has been performed at every British coronation since 1727. Almost blind, he died in 1759, a respected and rich man, and was given a state funeral at Westminster Abbey.

Handel composed more than forty opere serie over a period of more than thirty years. Since the late 1960s, interest in Handel's music has grown. The musicologist Winton Dean wrote that "Handel was not only a great composer; he was a dramatic genius of the first order." His music was admired by Classical-era composers, especially Mozart, Haydn and Beethoven.

Family

Handel was born in 1685 (the same year as Johann Sebastian Bach and Domenico Scarlatti) in Halle, in the Duchy of Magdeburg, then part of Brandenburg-Prussia. His parents were Georg Händel, aged 63, and Dorothea Taust. His father was an eminent barber-surgeon who served the court of Saxe-Weissenfels and the Margraviate of Brandenburg.

Halle was a relatively prosperous city, home of a salt-mining industry, a centre of trade, and a member of the Hanseatic League. The Margrave of Brandenburg became the administrator of the archepiscopal territories of Mainz, including Magdeburg when they converted, and by the early 17th century held his court in Halle, which attracted renowned musicians. Even the smaller churches all had "able organists and fair choirs", and humanities and the letters thrived (Shakespeare was performed in the theatres early in the 17th century). The Thirty Years' War brought extensive destruction to Halle, and by the 1680s it was impoverished. However, since the middle of the war the city had been under the administration of the Duke of Saxony, and soon after the end of the war he would bring musicians trained in Dresden to his court in Weissenfels.

The arts and music, however, flourished only among the higher strata (not only in Halle but throughout Germany), of which Handel's family was not a part. Georg Händel (senior) was born at the beginning of the war and was apprenticed to a barber in Halle at the age of 14 after his father died. When he was 20, he married the widow of the official barber-surgeon of a suburb of Halle, inheriting his practice. With this, Georg determinedly began the process of becoming self-made; by dint of his "conservative, steady, thrifty, unadventurous" lifestyle, he guided the five children he had with Anna who reached adulthood into the medical profession (except his youngest daughter, who married a government official). Anna died in 1682. Within a year Georg married again, this time to the daughter of a Lutheran minister, Pastor Georg Taust of the Church of St. Bartholomew in Giebichenstein, who himself came from a long line of Lutheran pastors. George Frideric was the second child of this marriage; the first son was stillborn. Two younger sisters arrived afterwards: Dorthea Sophia, born on 6 October 1687, and Johanna Christiana, born on 10 January 1690.

Early education
Early in his life Handel is reported to have attended the Gymnasium in Halle, where the headmaster, Johann Praetorius (musician), was reputed to be an ardent musician. Whether Handel remained there, and if he did for how long, is unknown, but many biographers suggest that he was withdrawn from school by his father, based on the characterization of him by Handel's first biographer, John Mainwaring. Mainwaring is the source for almost all information (little as it is) of Handel's childhood, and much of that information came from J. C. Smith Jr., Handel's confidant and copyist. Whether it came from Smith or elsewhere, Mainwaring frequently relates misinformation. It is from Mainwaring that the portrait comes of Handel's father as implacably opposed to any musical education. Mainwaring writes that Georg Händel was "alarmed" at Handel's very early propensity for music, "took every measure to oppose it", including forbidding any musical instrument in the house and preventing Handel from going to any house where they might be found. This did nothing to dampen young Handel's inclination; in fact, it did the reverse. Mainwaring tells the story of Handel's secret attic spinet: Handel "found means to get a little clavichord privately convey'd to a room at the top of the house. To this room he constantly stole when the family was asleep". Although both John Hawkins and Charles Burney credited this tale, Schoelcher found it nearly "incredible" and a feat of "poetic imagination" and Lang considers it one of the unproven "romantic stories" that surrounded Handel's childhood. But Handel had to have had some experience with the keyboard to have made the impression in Weissenfels that resulted in his receiving formal musical training.

Musical education
Sometime between the ages of seven and nine, Handel accompanied his father to Weissenfels, where he came under the notice of one whom Handel thereafter always regarded throughout life as his benefactor, Duke Johann Adolf I. Somehow Handel made his way to the court organ in the palace chapel of the Holy Trinity, where he surprised everyone with his playing. Overhearing this performance and noting the youth of the performer caused the Duke, whose suggestions were not to be disregarded, to recommend to Georg Händel that Handel be given musical instruction. Handel's father engaged the organist at the Halle parish church, the young Friedrich Wilhelm Zachow, to instruct Handel. Zachow would be the only teacher that Handel ever had. Because of his church employment, Zachow was an organist "of the old school", revelling in fugues, canons, and counterpoint. But he was also familiar with developments in music across Europe and his own compositions "embraced the new concerted, dramatic style".

When Zachow discovered the talent of Handel, he introduced him "to a vast collection of German and Italian music, which he possessed, sacred and profane, vocal and instrumental compositions of different schools, different styles, and of every master". Many traits considered "Handelian" can be traced back to Zachow's music. At the same time Handel continued practice on the harpsichord, and learned violin and organ, but according to Burney his special affection was for the hautbois (oboe). Schoelcher speculates that his youthful devotion to the instrument explains the large number of pieces he composed for the oboe.

With respect to instruction in composition, in addition to having Handel apply himself to traditional fugue and cantus firmus work, Zachow, recognising Handel's precocious talents, systematically introduced Handel to the variety of styles and masterworks contained in his extensive library. He did this by requiring Handel to copy selected scores. "I used to write like the devil in those days", Handel recalled much later.

Much of this copying was entered into a notebook that Handel maintained for the rest of his life. Although it has since disappeared, the notebook has been sufficiently described to understand what pieces Zachow wished Handel to study. Among the chief composers represented in this exercise book were Johann Krieger, an "old master" in the fugue and prominent organ composer, Johann Caspar Kerll, a representative of the "southern style" after his teacher Girolamo Frescobaldi and imitated later by Handel, Johann Jakob Froberger, an "internationalist" also closely studied by Buxtehude and Bach, and Georg Muffat, whose amalgam of French and Italian styles and his synthesis of musical forms influenced Handel.

Mainwaring writes that during this time Zachow had begun to have Handel assume some of his church duties. Zachow, Mainwaring asserts, was "often" absent, "from his love of company, and a cheerful glass", and Handel, therefore, performed on organ frequently. What is more, according to Mainwaring, Handel began composing, at the age of nine, church services for voice and instruments "and from that time actually did compose a service every week for three years successively". Mainwaring ends this chapter of Handel's life by concluding that three or four years had been enough to allow Handel to surpass Zachow, and Handel had become "impatient for another situation"; "Berlin was the place agreed upon." Carelessness with dates or sequences (and possibly imaginative interpretation by Mainwaring) makes this period confused.

After the death of Handel's father
Handel's father died on 11 February 1697. It was German custom for friends and family to compose funeral odes for a substantial burgher like Georg, and young Handel discharged his duty with a poem dated 18 February and signed with his name and (in deference to his father's wishes) "dedicated to the liberal arts." At the time Handel was studying either at Halle's Lutheran Gymnasium or the Latin School.

Mainwaring has Handel travelling to Berlin the next year, 1698. The problem with Mainwaring as an authority for this date, however, is that he tells of how Handel's father communicated with the "king" during Handel's stay, declining the Court's offer to send Handel to Italy on a stipend and that his father died "after his return from Berlin." But since Georg Händel died in 1697, either the date of the trip or Mainwaring's statements about Handel's father must be in error. Early biographers solved the problem by making the year of the trip 1696, then noting that at the age of 11, Handel would need a guardian, so they have Handel's father or a friend of the family accompany him, all the while puzzling over why the elder Handel, who wanted Handel to become a lawyer, would spend the sum to lead his son further into the temptation of music as a career. Schoelcher for example has Handel travelling to Berlin at 11, meeting both Bononcini and Attilio Ariosti in Berlin and then returning at the direction of his father. But Ariosti was not in Berlin before the death of Handel's father, and Handel could not have met Bononcini in Berlin before 1702. Modern biographers either accept the year as 1698, since most reliable older authorities agree with it, and discount what Mainwaring says about what took place during the trip or assume that Mainwaring conflated two or more visits to Berlin, as he did with Handel's later trips to Venice.

University
Perhaps to fulfil a promise to his father or simply because he saw himself as "dedicated to the liberal arts", on 10 February 1702 Handel matriculated at the University of Halle. That university had only recently been founded. In 1694, the Elector of Brandenburg Frederick III (later Prussian King Frederick I) created the school, largely to provide a lecture forum for the jurist Christian Thomasius who had been expelled from Leipzig for his liberal views. Handel did not enrol in the faculty of law, although he almost certainly attended lectures. Thomasius was an intellectual and academic crusader, who was the first German academic to lecture in German and also denounced witch trials. Lang believes that Thomasius instilled in Handel a "respect for the dignity and freedom of man's mind and the solemn majesty of the law", principles that would have drawn him to and kept him in England for half a century. Handel also there encountered theologian and professor of Oriental languages August Hermann Francke, who was particularly solicitous of children, especially orphans. The orphanage he founded became a model for Germany, and undoubtedly influenced Handel's own charitable impulse when he assigned the rights of Messiah to London's Foundling Hospital.

Shortly after commencing his university education, Handel (though Lutheran) on 13 March 1702 accepted the position of organist at the Calvinist Cathedral in Halle, the Domkirche, replacing J. C. Leporin, for whom he had acted as assistant. The position, which was a one-year probationary appointment, showed the foundation he had received from Zachow, for a church organist and cantor was a highly prestigious office. From it, he received 5 thalers a year and lodgings in the run-down castle of Moritzburg.

Around this same time, Handel made the acquaintance of Telemann. Four years Handel's senior, Telemann was studying law at Leipzig and was assisting cantor Johann Kuhnau (Bach's predecessor at the Thomaskirche there). Telemann recalled forty years later in an autobiography for Mattheson's Grundlage: "The writing of the excellent Johann Kuhnau served as a model for me in fugue and counterpoint; but in fashioning melodic movements and examining them Handel and I were constantly occupied, frequently visiting each other as well as writing letters."

Halle compositions
Although Mainwaring records that Handel wrote weekly when assistant to Zachow and as probationary organist at Domkirche part of his duty was to provide suitable music, no sacred compositions from his Halle period can now be identified. Mattheson, however, summarised his opinion of Handel's church cantatas written in Halle: "Handel in those days set very, very long arias and sheerly unending cantatas which, while not possessing the proper knack or correct taste, were perfect so far as harmony is concerned."

Early chamber works do exist, but it is difficult to date any of them to Handel's time in Halle. Many historians until recently followed Chrysander and designated the six trio sonatas for two oboes and basso continuo as his first known composition, supposedly written in 1696 (when Handel was 11). Lang doubts the dating based on a handwritten date of a copy (1700) and stylistic considerations. Lang writes that the works "show thorough acquaintance with the distilled sonata style of the Corelli school" and are notable for "the formal security and the cleanness of the texture." Hogwood considers all of the oboe trio sonatas spurious and even suggests that some parts cannot be performed on oboe. That authentic manuscript sources do not exist and that Handel never recycled any material from these works makes their authenticity doubtful. Other early chamber works were printed in Amsterdam in 1724 as opus 1, but it is impossible to tell which are early works in their original form, rather than later re-workings by Handel, a frequent practice of his.

From Hamburg to Italy

Handel's probationary appointment to Domkirche expired in March 1703. By July Handel was in Hamburg. Since he left no explanation for the move biographers have offered their own speculation.

Donald Burrows believes that the answer can be found by untangling Mainwaring's confused chronology of the trip to Berlin. Burrows dates this trip to 1702 or 1703 (after his father's death) and concludes that since Handel (through a "friend and relation" at the Berlin court) turned down Frederick's offer to subsidise his musical education in Italy (with the implicit understanding that he would become a court musician on his return), Handel was no longer able to expect preferment (whether as a musician, lawyer or otherwise) within Brandenburg-Prussia. Since he was attracted to secular, dramatic music (by meeting the Italians Bononcini and Attilio Ariosti and through the influence of Telemann), Hamburg, a free city with an established opera company, was the logical choice.

The question remains, however, why Handel rejected the King's offer, given that Italy was the centre of opera. Lang suggests that influenced by the teachings of Thomasius, Handel's character was such that he was unable to make himself subservient to anyone, even a king. Lang sees Handel as someone who could not accept class distinctions that required him to regard himself as a social inferior. "What Handel craved was personal freedom to raise himself out of his provincial milieu to a life of culture." Burrows notes that, like his father, Handel was able to accept royal (and aristocratic) favours without considering himself a court servant; and so, given the embarrassed financial condition of his mother, Handel set off for Hamburg to obtain experience while supporting himself.

In 1703, he accepted a position as violinist and harpsichordist in the orchestra of the Hamburg Oper am Gänsemarkt. There he met the composers Johann Mattheson, Christoph Graupner and Reinhard Keiser. Handel's first two operas, Almira and Nero, were produced in 1705. He produced two other operas, Daphne and Florindo, performed in 1708.

According to Mainwaring, in 1706 Handel travelled to Italy at the invitation of Ferdinando de' Medici. (Other sources say Handel was invited by Gian Gastone de' Medici, whom Handel had met in 1703–04 in Hamburg.) Ferdinando, who had a keen interest in opera, was trying to make Florence Italy's musical capital by attracting the leading talents of his day. In Italy, Handel met librettist Antonio Salvi, with whom he later collaborated. Handel left for Rome and since opera was (temporarily) banned in the Papal States, composed sacred music for the Roman clergy. His famous Dixit Dominus (1707) is from this era. He also composed cantatas in pastoral style for musical gatherings in the palaces of duchess Aurora Sanseverino (whom Mainwaring called "Donna Laura") one of the most influential patrons from the Kingdom of Naples, and cardinals Pietro Ottoboni, Benedetto Pamphili and Carlo Colonna. Two oratorios, La resurrezione and Il trionfo del tempo, were produced in a private setting for Ruspoli and Ottoboni in 1709 and 1710, respectively. Rodrigo, his first all-Italian opera, was produced in the Cocomero theatre in Florence in 1707. Agrippina was first produced in 1709 at Teatro San Giovanni Grisostomo in Venice, owned by the Grimanis. The opera, with a libretto by Cardinal Vincenzo Grimani, ran for 27 nights successively. The audience, thunderstruck with the grandeur and sublimity of his style, applauded for Il caro Sassone ("the dear Saxon" – referring to Handel's German origins).

Arrival
In June 1710, Handel became Kapellmeister to German prince George, the Elector of Hanover, but left at the end of the year. It is likely he was also invited by Charles Montagu the former ambassador in Venice to visit England. He visited Anna Maria Luisa de' Medici and her husband in Düsseldorf on his way to London. With his opera Rinaldo, based on La Gerusalemme Liberata by the Italian poet Torquato Tasso, Handel enjoyed great success, although it was composed quickly, with many borrowings from his older Italian works. This work contains one of Handel's favourite arias, Cara sposa, amante cara, and the famous Lascia ch'io pianga.

Handel went back to Halle twice, to attend the wedding of his sister and the baptism of her daughter, but decided to settle permanently in England in 1712. In the summer of 1713, he lived at Mr. Mathew Andrews' estate in Barn Elms, Surrey. He received a yearly income of £200 from Queen Anne after composing for her the Utrecht Te Deum and Jubilate, first performed in 1713.

One of his most important patrons was the 3rd Earl of Burlington and 4th Earl of Cork, a young and extremely wealthy member of an Anglo-Irish aristocratic family. While living in the mansion of Lord Burlington, Handel wrote Amadigi di Gaula, a "magic" opera, about a damsel in distress, based on the tragedy by Antoine Houdar de la Motte.

The conception of an opera as a coherent structure was slow to capture Handel's imagination and he composed no operas for five years. In July 1717, Handel's Water Music was performed more than three times on the River Thames for King George I and his guests. It is said the compositions spurred reconciliation between Handel and the king, supposedly annoyed by the composer's abandonment of his Hanover post.

At Cannons (1717–19)
In 1717, Handel became house composer at Cannons in Middlesex, where he laid the cornerstone for his future choral compositions in the Chandos Anthems. Romain Rolland wrote that these anthems (or Psalms) stood in relation to Handel's oratorios, much the same way that the Italian cantatas stood to his operas: "splendid sketches of the more monumental works." Another work, which he wrote for The 1st Duke of Chandos, the owner of Cannons, was Acis and Galatea: during Handel's lifetime, it was his most performed work. Winton Dean wrote that "the music catches breath and disturbs the memory".

In 1719, the Duke of Chandos became one of the composer's important patrons and a primary subscriber to his new opera company, the Royal Academy of Music, though his patronage declined after Chandos lost large sums of money in the South Sea Bubble, which burst in 1720 in one of history's greatest financial cataclysms. Handel himself invested in the South Sea Company in 1716, when its share prices were low and sold them before the "bubble" burst in 1720. In 1720, Handel invested in the slave-trading Royal African Company (RAC), following in the steps of his patron (the Duke of Chandos was one of the leading investors in the RAC). As noted by music historian David Hunter, 32 per cent of the subscribers and investors in the Royal Academy of Music, or their close family members, held investments in the RAC as well.

Royal Academy of Music (1719–34)
In May 1719, The 1st Duke of Newcastle, the Lord Chamberlain, ordered Handel to look for new singers. Handel travelled to Dresden to attend the newly built opera. He saw Teofane by Antonio Lotti, and engaged members of the cast for the Royal Academy of Music, founded by a group of aristocrats to assure themselves a constant supply of baroque opera or opera seria. Handel may have invited John Smith, his fellow student in Halle, and his son Johann Christoph Schmidt, to become his secretary and amanuensis. By 1723 he had moved into a Georgian house at 25 Brook Street, which he rented for the rest of his life. This house, where he rehearsed, copied music, and sold tickets, is now the Handel House Museum. During twelve months between 1724 and 1725, Handel wrote three successful operas, Giulio Cesare, Tamerlano and Rodelinda. Handel's operas are filled with da capo arias, such as Svegliatevi nel core. After composing Silete venti, he concentrated on opera and stopped writing cantatas. Scipio, from which the regimental slow march of the British Grenadier Guards is derived, was performed as a stopgap, waiting for the arrival of Faustina Bordoni.

In 1727, Handel was commissioned to write four anthems for the Coronation ceremony of King George II. One of these, Zadok the Priest, has been played at every British coronation ceremony since. The words to Zadok the Priest are taken from the King James Bible. In 1728, John Gay's The Beggar's Opera, which made fun of the type of Italian opera Handel had popularised in London, premiered at Lincoln's Inn Fields Theatre and ran for 62 consecutive performances, the longest run in theatre history up to that time. After nine years the Royal Academy of Music ceased to function but Handel soon started a new company.

The Queen's Theatre at the Haymarket (now His Majesty's Theatre), established in 1705 by architect and playwright John Vanbrugh, quickly became an opera house. Between 1711 and 1739, more than 25 of Handel's operas premièred there. In 1729, Handel became joint manager of the theatre with John James Heidegger.

Handel travelled to Italy to engage new singers and also composed seven more operas, among them the comic masterpiece Partenope and the "magic" opera Orlando. After two commercially successful English oratorios Esther and Deborah, he was able to invest again in the South Sea Company. Handel reworked his Acis and Galatea which then became his most successful work ever. Handel failed to compete with the Opera of the Nobility, who engaged musicians such as Johann Adolph Hasse, Nicolo Porpora and the famous castrato Farinelli. The strong support by Frederick, Prince of Wales caused conflicts in the royal family. In March 1734 Handel composed a wedding anthem This is the day which the Lord hath made, and a serenata Parnasso in Festa for Anne, Princess Royal.

Despite the problems the Opera of the Nobility was causing him at the time, Handel's neighbour in Brook Street, Mary Delany, reported on a party she invited Handel to at her house on 12 April 1734 where he was in good spirits:"I had Lady Rich and her daughter, Lady Cath. Hanmer and her husband, Mr. and Mrs. Percival, Sir John Stanley and my brother, Mrs. Donellan, Strada [star soprano of Handel's operas] and Mr. Coot. Lord Shaftesbury begged of Mr. Percival to bring him, and being a profess'd friend of Mr. Handel (who was here also) was admitted; I never was so well entertained at an opera! Mr. Handel was in the best humour in the world, and played lessons and accompanied Strada and all the ladies that sang from seven o'clock till eleven. I gave them tea and coffee, and about half an hour after nine had a salver brought in of chocolate, mulled white wine, and biscuits. Everybody was easy and seemed pleased."

Opera at Covent Garden (1734–41)
In 1733, the Earl of Essex received a letter with the following sentence: "Handel became so arbitrary a prince, that the Town murmurs." The board of chief investors expected Handel to retire when his contract ended, but Handel immediately looked for another theatre. In cooperation with John Rich he started his third company at Covent Garden Theatre. Rich was renowned for his spectacular productions. He suggested Handel use his small chorus and introduce the dancing of Marie Sallé, for whom Handel composed Terpsicore. In 1735, he introduced organ concertos between the acts. For the first time, Handel allowed Gioacchino Conti, who had no time to learn his part, to substitute arias. Financially, Ariodante was a failure, although he introduced ballet suites at the end of each act. Alcina, his last opera with a magic content, and Alexander's Feast or the Power of Music based on John Dryden's Alexander's Feast starred Anna Maria Strada del Pò and John Beard.

Early 1737 he produced Arminio and Giustino, completed Berenice, revived Partenope, and continued with Il Parnasso in Festa, Alexander's Feast, and the revised The Triumph of Time and Truth which premiered on 23 March. In April Handel suffered a mild stroke, or rheumatic palsy, resulting in temporary paralysis in his right hand and arm. After brief signs of a recovery, he had a relapse in May, with an accompanying deterioration in his mental capacities. He had strong competition from John Frederick Lampe; The Dragon of Wantley was first performed at the Little Theatre in the Haymarket in London on 16 May 1737. It was a parody of the Italian opera seria.

In Autumn 1737 the fatigued Handel reluctantly followed the advice of his physicians and went to take the cure in the spa towns of Royal Tunbridge Wells, Aix-la-Chapelle (Burtscheid) in September. All the symptoms of his "disorder" vanished by November. On Christmas Eve Handel finished the score of Faramondo, but its composition was interrupted by that of the Funeral Anthem for Queen Caroline. On Boxing Day he began the composition of Serse, the only comic opera that Handel ever wrote and worked with Elisabeth Duparc.

A harp and organ concerto (HWV 294) and Alexander's Feast were published in 1738 by John Walsh. He composed music for a musical clock with a pipe organ built by Charles Clay; it was bought by Gerrit Braamcamp and was in 2016 acquired by the Museum Speelklok in Utrecht. Deidamia, his last opera, a co-production with the Earl of Holderness, was performed three times in 1741. Handel gave up the opera business, while he enjoyed more success with his English oratorios.

Oratorio
Il trionfo del tempo e del disinganno, an allegory, Handel's first oratorio was composed in Italy in 1707, followed by La resurrezione in 1708 which uses material from the Bible. The circumstances of Esther and its first performance, possibly in 1718, are obscure. Another 12 years had passed when an act of piracy caused him to take up Esther once again. Three earlier performances aroused such interest that they naturally prompted the idea of introducing it to a larger public. Next came Deborah, strongly coloured by the coronation anthems and Athaliah, his third English Oratorio. In these three oratorios Handel laid the foundation for the traditional use of the chorus which marks his later oratorios. Handel became sure of himself, broader in his presentation, and more diverse in his composition.

It is evident how much he learned from Arcangelo Corelli about writing for instruments, and from Alessandro Scarlatti about writing for the solo voice; but there is no single composer who taught him how to write for chorus. Handel tended more and more to replace Italian soloists with English ones. The most significant reason for this change was the dwindling financial returns from his operas. Thus a tradition was created for oratorios which was to govern their future performance. The performances were given without costumes and action; the singers appeared in their own clothes.

In 1736, Handel produced Alexander's Feast. John Beard appeared for the first time as one of Handel's principal singers and became Handel's permanent tenor soloist for the rest of Handel's life. The piece was a great success and it encouraged Handel to make the transition from writing Italian operas to English choral works. In Saul, Handel was collaborating with Charles Jennens and experimenting with three trombones, a carillon and extra-large military kettledrums (from the Tower of London), to be sure "...it will be most excessive noisy". Saul and Israel in Egypt, both from 1739, head the list of great, mature oratorios, in which the da capo aria became the exception and not the rule. Israel in Egypt consists of little else but choruses, borrowing from the Funeral Anthem for Queen Caroline. In his next works, Handel changed his course. In these works he laid greater stress on the effects of orchestra and soloists; the chorus retired into the background. L'Allegro, il Penseroso ed il Moderato has a rather diverting character; the work is light and fresh.

During the summer of 1741, the 3rd Duke of Devonshire invited Handel to Dublin, capital of the Kingdom of Ireland, to give concerts for the benefit of local hospitals. His Messiah was first performed at the New Music Hall in Fishamble Street on 13 April 1742, with 26 boys and five men from the combined choirs of St Patrick's and Christ Church cathedrals participating. Handel secured a balance between soloists and chorus which he never surpassed. In 1747, Handel wrote his oratorio Alexander Balus. This work was produced at Covent Garden Theatre in London, on 23 March 1748, and to the aria ''Hark! hark! He strikes the golden lyre'', Handel wrote the accompaniment for mandolin, harp, violin, viola, and violoncello. Another of his English oratorios, Solomon, was first performed on 17 March 1749 at the Covent Garden Theatre. Solomon contains a short and lively instrumental passage for two oboes and strings in act 3, known as "The Arrival of the Queen of Sheba".

The use of English soloists reached its height at the first performance of Samson. The work is highly theatrical. The role of the chorus became increasingly important in his later oratorios. Jephtha was first performed on 26 February 1752; even though it was his last oratorio, it was no less a masterpiece than his earlier works.

Later years

In 1749, Handel composed Music for the Royal Fireworks; 12,000 people attended the first performance. In 1750, he arranged a performance of Messiah to benefit the Foundling Hospital, a children's home in London. The performance was considered a great success and was followed by annual concerts that continued throughout his life. In recognition of his patronage, Handel was made a governor of the Hospital the day after his initial concert. He bequeathed a copy of Messiah to the institution upon his death. His involvement with the Foundling Hospital is today commemorated with a permanent exhibition in London's Foundling Museum, which also holds the Gerald Coke Handel Collection. In addition to the Foundling Hospital, Handel also gave to a charity that assisted impoverished musicians and their families.

In August 1750, on a journey back from Germany to London, Handel was seriously injured in a carriage accident between The Hague and Haarlem in the Netherlands. In 1751, one eye started to fail. The cause was a cataract which was operated on by the great charlatan Chevalier Taylor. This did not improve his eyesight and possibly made it worse. He was completely blind by 1752. He died in 1759 at home in Brook Street, at the age of 74. The last performance he attended was of Messiah. Handel was buried in Westminster Abbey. More than three thousand mourners attended his funeral, which was given full state honours.

Handel never married and kept his personal life private. His initial will bequeathed the bulk of his estate to his niece Johanna, but four codicils distributed much of his estate to other relations, servants, friends and charities.

Handel owned an art collection that was auctioned posthumously in 1760. The auction catalogue listed approximately seventy paintings and ten prints (other paintings were bequeathed).

Overview
Handel's compositions include 42 operas, 24 oratorios, more than 120 cantatas, trios and duets, numerous arias, odes and serenatas, solo and trio sonatas, 18 concerti grossi, and 12 organ concertos. His most famous work, the oratorio Messiah with its "Hallelujah" chorus, is among the most popular works in choral music. The Lobkowicz Palace in Prague holds Mozart's copy of Messiah, complete with handwritten annotations. Among the works with opus numbers published and popularised in his lifetime are the Organ concertos Op. 4 and Op. 7, together with the Opus 3 and Opus 6 Concerti grossi; the latter incorporates an earlier organ concerto, The Cuckoo and the Nightingale, in which birdsong is imitated in the upper registers of the organ. Also notable are his 16 keyboard suites, especially The Harmonious Blacksmith.

Catalogues
The first published catalogue of Handel's works appeared as an appendix to Mainwaring's Memoirs. Between 1787 and 1797 Samuel Arnold compiled a 180-volume collection of Handel's works—however, it was far from complete. Also incomplete was the collection produced between 1843 and 1858 by the English Handel Society (founded by Sir George Macfarren).

The 105-volume Händel-Gesellschaft ("Handel Society") edition was published between 1858 and 1902 – mainly due to the efforts of Friedrich Chrysander. For modern performance, the realisation of the basso continuo reflects 19th-century practice. Vocal scores drawn from the edition were published by Novello in London, but some scores, such as the vocal score to Samson, are incomplete.

The continuing Hallische Händel-Ausgabe edition was first inaugurated in 1955 in the Halle region in Saxony-Anhalt, East Germany. It did not start as a critical edition, but after heavy criticism of the first volumes, which were performing editions without a critical apparatus (for example, the opera Serse was published with the title character recast as a tenor, reflecting pre-war German practice), it repositioned itself as a critical edition. Influenced in part by cold-war realities, editorial work was inconsistent: misprints were found in abundance and editors failed to consult important sources. In 1985, a committee was formed to establish better standards for the edition. The reunification of Germany in 1990 removed communication problems, and the volumes issued have since shown a significant improvement in standards.

Between 1978 and 1986 the German academic Bernd Baselt catalogued Handel's works in his Händel-Werke-Verzeichnis publication. The catalogue has achieved wide acceptance and is used as the modern numbering system, with each of Handel's works designated an "HWV" number – for example, Messiah is catalogued as "HWV 56".

Legacy
Handel's works were collected and preserved by two men: Sir Samuel Hellier, a country squire whose musical acquisitions form the nucleus of the Shaw–Hellier Collection, and the abolitionist Granville Sharp. The catalogue accompanying the National Portrait Gallery exhibition marking the tercentenary of the composer's birth calls them two men of the late eighteenth century "who have left us solid evidence of the means by which they indulged their enthusiasm". With his English oratorios, such as Messiah and Solomon, the coronation anthems, and other works including Water Music and Music for the Royal Fireworks, Handel became a national icon in Britain, and featured in the BBC series The Birth of British Music: Handel – The Conquering Hero.

After his death, Handel's Italian operas fell into obscurity, except for selections such as the aria from Serse, "Ombra mai fu". The oratorios continued to be performed but not long after Handel's death they were thought to need some modernisation, and Mozart orchestrated German versions of Messiah and other works. Throughout the 19th century and first half of the 20th century, particularly in the Anglophone countries, his reputation rested primarily on his English oratorios, which were customarily performed by choruses of amateur singers on solemn occasions. The centenary of his death, in 1859, was celebrated by a performance of Messiah at The Crystal Palace, involving 2,765 singers and 460 instrumentalists, who played for an audience of about 10,000 people.

Recent decades have revived his secular cantatas and what one might call 'secular oratorios' or 'concert operas'. Of the former, Ode for St. Cecilia's Day (1739) (set to texts by John Dryden) and Ode for the Birthday of Queen Anne (1713) are noteworthy. For his secular oratorios, Handel turned to classical mythology for subjects, producing such works as Acis and Galatea (1719), Hercules (1745) and Semele (1744). These works have a close kinship with the sacred oratorios, particularly in the vocal writing for the English-language texts. They also share the lyrical and dramatic qualities of Handel's Italian operas. As such, they are sometimes fully staged as operas. With the rediscovery of his theatrical works, Handel, in addition to his renown as instrumentalist, orchestral writer, and melodist, is now perceived as being one of opera's great musical dramatists.

Reception
Handel has generally been accorded high esteem by fellow composers, both in his own time and since. Johann Sebastian Bach attempted, unsuccessfully, to meet Handel while he was visiting Halle. (Handel was born in the same year as Bach and Domenico Scarlatti.) Mozart is reputed to have said of him, "Handel understands affect better than any of us. When he chooses, he strikes like a thunder bolt." To Beethoven he was "the master of us all... the greatest composer that ever lived. I would uncover my head and kneel before his tomb." Beethoven emphasised above all the simplicity and popular appeal of Handel's music when he said, "Go to him to learn how to achieve great effects, by such simple means."

Borrowings
Since 1831, when William Crotch raised the issue in his Substance of Several Lectures on Music, scholars have extensively studied Handel's "borrowing" of music from other composers. Summarising the field in 2005, Richard Taruskin wrote that Handel "seems to have been the champion of all parodists, adapting both his own works and those of other composers in unparalleled numbers and with unparalleled exactitude." Among the composers whose music has been shown to have been re-used by Handel are Alessandro Stradella, Gottlieb Muffat, Alessandro Scarlatti, Domenico Scarlatti Giacomo Carissimi, Georg Philipp Telemann, Carl Heinrich Graun, Leonardo Vinci, Jacobus Gallus, Francesco Antonio Urio, Reinhard Keiser, Francesco Gasparini, Giovanni Bononcini, William Boyce, Henry Lawes, Michael Wise, Agostino Steffani, Franz Johann Habermann, and numerous others.

In an essay published in 1985, John H. Roberts demonstrated that Handel's borrowings were unusually frequent even for his own era, enough to have been criticised by contemporaries (notably Johann Mattheson); Roberts suggested several reasons for Handel's practice, including Handel's attempts to make certain works sound more up-to-date and more radically, his "basic lack of facility in inventing original ideas" – though Roberts took care to argue that this does not "diminish Handel's stature", which should be "judged not by his methods, still less by his motives in employing them, but solely by the effects he achieves."

Homages
After Handel's death, many composers wrote works based on or inspired by his music. The first movement from Louis Spohr's Symphony No. 6, Op. 116, "The Age of Bach and Handel", resembles two melodies from Handel's Messiah. In 1797, Ludwig van Beethoven published the 12 Variations in G major on "See the conqu’ring hero comes" from Judas Maccabaeus by Handel, for cello and piano. In 1822, Beethoven composed the overture The Consecration of the House, which also bears the influence of Handel. Guitar virtuoso Mauro Giuliani composed his Variations on a Theme by Handel, Op. 107 for guitar, based on Handel's Suite No. 5 in E major, HWV 430, for harpsichord.

In 1861, using a theme from the second of Handel's harpsichord suites, Johannes Brahms wrote the Variations and Fugue on a Theme by Handel, Op. 24, one of his most successful works (praised by Richard Wagner). Several works by the French composer Félix-Alexandre Guilmant use Handel's themes; for example, his March on a Theme by Handel uses a theme from Messiah. French composer and flautist Philippe Gaubert wrote his Petite marche for flute and piano based on the fourth movement of Handel's Trio Sonata, Op. 5, No. 2, HWV 397. Argentine composer Luis Gianneo composed his Variations on a Theme by Handel for piano. In 1911, Australian-born composer and pianist Percy Grainger based one of his most famous works on the final movement of Handel's Suite No. 5 in E major (just like Giuliani). He first wrote some variations on the theme, which he titled Variations on Handel's 'The Harmonious Blacksmith' . Then he used the first sixteen bars of his set of variations to create Handel in the Strand, one of his most beloved pieces, of which he made several versions (for example, the piano solo version from 1930). Arnold Schoenberg's Concerto for String Quartet and Orchestra in B-flat major (1933) was composed after Handel's Concerto Grosso, Op. 6/7.

Veneration
In the Lutheran Calendar of Saints Handel and Bach share the date 28 July with Heinrich Schütz, and Handel and Bach are commemorated in the calendar of saints prepared by the Order of Saint Luke for the use of the United Methodist Church. The Book of Common Worship of the Presbyterian Church (USA) (Westminster John Knox Press, 2018) commemorates him on 20 April.

Fictional depictions
In 1942, Handel was the subject of the British biographical film The Great Mr. Handel directed by Norman Walker and starring Wilfrid Lawson. It was made at Denham Studios by the Rank Organisation, and shot in Technicolor. He is also the central character in the television films God Rot Tunbridge Wells! (1985) and Handel's Last Chance (1996) and the stage play All the Angels (2015). Handel was portrayed by Jeroen Krabbé as the antagonist in the film Farinelli (1994).

Scores and recordings

 * : includes Complete Works Edition (Ausgabe der Deutschen Händelgesellschaft)
 * The Mutopia Project provides free downloading of sheet music and MIDI files for some of Handel's works.
 * Digitized images of Old English Songs, containing works by Handel, housed at the University of Kentucky Libraries Special Collections
 * George Frideric Handel cylinder recordings, from the UCSB Cylinder Audio Archive at the University of California, Santa Barbara Library.
 * KunstDerFuge .mid files: George Frideric Handel – MIDI files
 * KunstDerFuge .mid files: George Frideric Handel – MIDI files

title: George Frideric Handel


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Giovanni Pierluigi da Palestrina

Giovanni Pierluigi da Palestrina (between 3 February 1525 and 2 February 1526 – 2 February 1594) was an Italian composer of late Renaissance music. The central representative of the Roman School, with Orlande de Lassus and Tomás Luis de Victoria, Palestrina is considered the leading composer of late 16th-century Europe.

Primarily known for his masses and motets, which number over 105 and 250 respectively, Palestrina had a long-lasting influence on the development of church and secular music in Europe, especially on the development of counterpoint. According to Grove Music Online, Palestrina's "success in reconciling the functional and aesthetic aims of Catholic church music in the post-Tridentine period earned him an enduring reputation as the ideal Catholic composer, as well as giving his style (or, more precisely, later generations' selective view of it) an iconic stature as a model of perfect achievement."

Biography
Palestrina was born in the town of Palestrina, near Rome, then part of the Papal States, to Neapolitan parents, Santo and Palma Pierluigi, in 1525, possibly on 3 February. His mother died on 16 January 1536, when Palestrina was 10. Documents suggest that he first visited Rome in 1537, when he was listed as a chorister at the Basilica of Santa Maria Maggiore, one of the papal basilicas of the Diocese of Rome, which allowed him to learn literature and music. In 1540, he moved to Rome, where he studied in the school of the Huguenot Claude Goudimel. He also studied with Robin Mallapert and Firmin Lebel. He spent most of his career in the city.

Palestrina came of age as a musician under the influence of the northern European style of polyphony, which owed its dominance in Italy primarily to two influential Netherlandish composers, Guillaume Du Fay and Josquin des Prez, who had spent significant portions of their careers there. Italy itself had yet to produce anyone of comparable fame or skill in polyphony. Orlando di Lasso, who accompanied Palestrina in his early years, also played an important role in the formation of his style as an adviser.

From 1544 to 1551, Palestrina was the organist of the Cathedral of St. Agapito, the principal church of his native city. In 1551 Pope Julius III (previously the Bishop of Palestrina) appointed Palestrina maestro di cappella or musical director of the Cappella Giulia, (Julian Chapel, in the sense of choir), the choir of the chapter of canons at St. Peter's Basilica. Palestrina dedicated to Julius III his first published compositions (1554), a book of Masses. It was the first book of Masses by a native composer, since in the Italian states of Palestrina's day, most composers of sacred music were from the Low Countries, France, or Spain. In fact the book was modelled on one by Cristóbal de Morales: the woodcut in the front is almost an exact copy of the one from the book by the Spanish composer.

In 1555, Pope Paul IV ordered that all papal choristers should be clerical. As Palestrina married early in life and had four children, he was unable to continue in the chapel as a layman.

During the next decade, Palestrina held positions similar to his Julian Chapel appointment at other chapels and churches in Rome, notably St. John Lateran (1555–1560, a post previously held by Lassus), and Santa Maria Maggiore (1561–1566). In 1571 he returned to the Julian Chapel and remained at St Peter's for the rest of his life. The decade of the 1570s was difficult for him personally: he lost his brother, two of his sons, and his wife, Lucrezia Gori, in three separate outbreaks of the plague (1572, 1575, and 1580, respectively). He seems to have considered becoming a priest at this time, but instead he remarried, this time to a wealthy widow, Virginia Dormoli. This finally gave him financial independence (he was not paid well as choirmaster) and was able to compose prolifically until his death.

He died in Rome of pleurisy on 2 February 1594. As was usual, Palestrina was buried on the same day he died, in a plain coffin with a lead plate on which was inscribed Ioannes Petrus Aloysius Praenestinus Musicae Princeps. A five-part Libera me Domine psalm for three choirs was sung at the funeral. Palestrina's funeral was held at St. Peter's, and he was buried beneath the floor of the basilica. His tomb was later covered by new construction and attempts to locate his grave have been unsuccessful.

Italian composers Giovanni Maria Nanino and Gregorio Allegri, both of them disciples of his school, continued his works.

Overview
Palestrina left hundreds of compositions, including 105 masses, 68 offertories, at least 140 madrigals and more than 300 motets. In addition, there are at least 72 hymns, 35 magnificats, 11 litanies, and four or five sets of lamentations. The Gloria melody from Palestrina's Magnificat Tertii Toni (1591) is widely used today in the resurrection hymn tune, Victory (The Strife Is O'er).

His attitude toward madrigals was somewhat enigmatic: whereas in the preface to his collection of Canticum canticorum (Song of Songs) motets (1584) he renounced the setting of profane texts, only two years later he was back in print with Book II of his secular madrigals (some of these being among the finest compositions in the medium). He published just two collections of madrigals with profane texts, one in 1555 and another in 1586. The other two collections were spiritual madrigals, a genre beloved by the proponents of the Counter-Reformation.

Palestrina's masses show how his compositional style developed over time. His Missa sine nomine seems to have been particularly attractive to Johann Sebastian Bach, who studied and performed it while writing the Mass in B minor. Most of Palestrina's masses appeared in thirteen volumes printed between 1554 and 1601, the last seven published after his death. One of his most important works, the Missa Papae Marcelli (Pope Marcellus Mass) has been historically associated with erroneous information involving the Council of Trent. According to this tale (which forms the basis of Hans Pfitzner's opera Palestrina), it was composed in order to persuade the Council of Trent that a draconian ban on the polyphonic treatment of text in sacred music (as opposed, that is, to a more directly intelligible homophonic treatment) was unnecessary. However, more recent scholarship shows that this mass was in fact composed before the cardinals convened to discuss the ban (possibly as much as 10 years before). Historical data indicates that the Council of Trent, as an official body, never actually banned any church music and failed to make any ruling or official statement on the subject. These stories originated from the unofficial points-of-view of some Council attendees who discussed their ideas with those not privy to the Council's deliberations. Those opinions and rumors have, over centuries, been transmuted into fictional accounts, put into print, and often incorrectly taught as historical fact. While Palestrina's compositional motivations are not known, he may have been quite conscious of the need for intelligible text; however, this was not to conform with any doctrine of the Counter-Reformation, because no such doctrine exists. His characteristic style remained consistent from the 1560s until the end of his life. Roche's hypothesis that Palestrina's seemingly dispassionate approach to expressive or emotive texts could have resulted from his having to produce many to order, or from a deliberate decision that any intensity of expression was unbecoming in church music, reflects modern expectations about expressive freedom and underestimates the extent to which the mood of Palestrina's settings is adapted to the liturgical occasions for which the texts were set, rather than the line-by-line meaning of the text, and depends on the distinctive characters of the church modes and variations in vocal grouping for expressive effect. Performing editions and recordings of Palestrina have tended to favour his works in the more familiar modes and standard (SATB) voicings, under-representing the expressive variety of his settings.

There are two comprehensive editions of Palestrina's works: a 33-volume edition published by Breitkopf and Härtel, in Leipzig Germany between 1862 and 1894 edited by Franz Xaver Haberl, and a 34-volume edition published in the mid twentieth century, by Fratelli Scalera, in Rome, Italy edited by R. Casimiri and others.

The "Palestrina Style"
One of the hallmarks of Palestrina's music is that dissonances are typically relegated to the "weak" beats in a measure. This produced a smoother and more consonant type of polyphony which is now considered to be definitive of late Renaissance music, given Palestrina's position as Europe's leading composer in the wake of Josquin des Prez (d. 1521).

The "Palestrina style" taught in college courses covering Renaissance counterpoint is often based on the codification by the 18th-century composer and theorist Johann Joseph Fux, published as Gradus ad Parnassum (Steps to Parnassus, 1725). Citing Palestrina as his model, Fux divided counterpoint into five species (hence the term "species counterpoint"), designed as exercises for the student, which deployed progressively more elaborate rhythmic combinations of voices while adhering to strict harmonic and melodic requirements. The method was widely adopted and was the main basis of contrapuntal training in the 19th century, but Fux had introduced a number of simplifications to the Palestrina style, notably the obligatory use of a cantus firmus in semibreves, which were corrected by later authors such as Knud Jeppesen and R. O. Morris. Palestrina's music conforms in many ways to Fux's rules, particularly in the fifth species but does not fit his pedagogical format.

The main insight, that the "pure" style of polyphony achieved by Palestrina followed an invariable set of stylistic and combinational requirements, was justified. Fux's manual was endorsed by his contemporary J.S. Bach, who himself arranged two of Palestrina's masses for performance.

According to Fux, Palestrina had established and followed these basic guidelines:

 * The flow of music is dynamic, not rigid or static.
 * Melody should contain few leaps between notes. (Jeppesen: "The line is the starting point of Palestrina's style".)
 * If a leap occurs, it must be small and immediately countered by stepwise motion in the opposite direction.
 * Dissonances are to be confined to suspensions, passing notes and weak beats. If one falls on a strong beat (in a suspension) it must be immediately resolved.

Fux omits to mention the manner in which the musical phrasing of Palestrina followed the syntax of the sentences he was setting to music, something not always observed by earlier composers. Also to be noticed in Palestrina is a great deal of tone painting. Elementary examples of this are descending musical motion with Latin words like descendit (descends) or of a static musical or cadential moment with the words de coelis (from heaven).

Reputation
Palestrina was extremely famous in his day, and if anything, his reputation and influence increased after his death. J.S. Bach studied and hand-copied Palestrina's first book of Masses, and in 1742 wrote his own adaptation of the Kyrie and Gloria of the Missa sine nomine. Felix Mendelssohn placed him in the pantheon of the greatest musicians, writing, "I always get upset when some praise only Beethoven, others only Palestrina and still others only Mozart or Bach. All four of them, I say, or none at all.".

Conservative music of the Roman school continued to be written in Palestrina's style (which in the 17th century came to be known as the prima prattica) by such students of his as Giovanni Maria Nanino, Ruggiero Giovanelli, Arcangelo Crivelli, Teofilo Gargari, Francesco Soriano, and Gregorio Allegri. As late as the 1750s, Palestrina's style was still the reference for composers working in the motet form, as can be seen by Francesco Barsanti's Sei Antifones 'in the style of Palestrina' (c. 1750; published by [Peter] Welcker, c. 1762).

Much research on Palestrina was done in the 19th century by Giuseppe Baini, who published a monograph in 1828 which made Palestrina famous again and reinforced the already existing legend that he was the "Saviour of Church Music" during the reforms of the Council of Trent.

20th and 21st century scholarship by and large retains the view that Palestrina was a strong and refined composer whose music represents a summit of technical perfection. Contemporary analysis highlighted the modern qualities in the compositions of Palestrina such as research of color and sonority, use of sonic grouping in large-scale setting, interest in vertical as well as horizontal organization, studied attention to text setting. These unique characteristics, together with effortless delivery and an indefinable "otherness", constitute to this day the attraction of Palestrina's work.

The Cagliari music conservatory in Cagliari, Italy is named in his honor.

In 2009 a film about the composer was produced by German television ZDF/Arte. Title: Palestrina - Prince of Music, directed by Georg Brintrup.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Giovanni Pierluigi da Palestrina


text:
Group velocity

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave. The expanding ring of waves is the wave group or wave packet, within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group.

History
The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877.

Definition and interpretation
The group velocity $v_{g}$ is defined by the equation:
 * $$v_{\rm g} \ \equiv\ \frac{\partial \omega}{\partial k}\,$$

where $ω$ is the wave's angular frequency (usually expressed in radians per second), and $k$ is the angular wavenumber (usually expressed in radians per meter). The phase velocity is: $v_{p} = ω/k$.

The function $ω(k)$, which gives $ω$ as a function of $k$, is known as the dispersion relation.

 * If $ω$ is directly proportional to  $k$, then the group velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity.
 * If ω is a linear function of k, but not directly proportional $(ω = ak + b)$, then the group velocity and phase velocity are different. The envelope of a wave packet (see figure on right) will travel at the group velocity, while the individual peaks and troughs within the envelope will move at the phase velocity.
 * If $ω$ is not a linear function of $k$, the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies (and hence different values of $k$), the group velocity $∂ω/∂k$ will be different for different values of $k$. Therefore, the envelope does not move at a single velocity, but its wavenumber components ($k$) move at different velocities, distorting the envelope. If the wavepacket has a narrow range of frequencies, and $ω(k)$ is approximately linear over that narrow range, the pulse distortion will be small, in relation to the small nonlinearity. See further discussion below. For example, for deep water gravity waves, $\omega = \sqrt{gk}$, and hence $v_{g} = v_{p} /2$. This underlies the Kelvin wake pattern for the bow wave of all ships and swimming objects. Regardless of how fast they are moving, as long as their velocity is constant, on each side the wake forms an angle of 19.47° = arcsin(1/3) with the line of travel.

Derivation
One derivation of the formula for group velocity is as follows.

Consider a wave packet as a function of position $x$ and time $t: α(x,t)$.

Let $A(k)$ be its Fourier transform at time $t = 0$,
 * $$ \alpha(x, 0) = \int_{-\infty}^\infty dk \, A(k) e^{ikx}.$$

By the superposition principle, the wavepacket at any time $t$ is
 * $$ \alpha(x, t) = \int_{-\infty}^\infty dk \, A(k) e^{i(kx - \omega t)},$$

where $ω$ is implicitly a function of  $k$.

Assume that the wave packet $α$ is almost monochromatic, so that  $A(k)$ is sharply peaked around a central wavenumber  $k_{0}$.

Then, linearization gives
 * $$\omega(k) \approx \omega_0 + \left(k - k_0\right)\omega'_0$$

where
 * $$\omega_0 = \omega(k_0)$$ and $$\omega'_0 = \left.\frac{\partial \omega(k)}{\partial k}\right|_{k=k_0}$$

(see next section for discussion of this step). Then, after some algebra,
 * $$ \alpha(x,t) = e^{i\left(k_0 x - \omega_0 t\right)}\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}.$$

There are two factors in this expression. The first factor, $$e^{i\left(k_0 x - \omega_0 t\right)}$$, describes a perfect monochromatic wave with wavevector $k_{0}$, with peaks and troughs moving at the phase velocity $$\omega_0/k_0$$  within the envelope of the wavepacket.

The other factor,
 * $$\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}$$,

gives the envelope of the wavepacket. This envelope function depends on position and time only through the combination $$(x - \omega'_0 t)$$.

Therefore, the envelope of the wavepacket travels at velocity
 * $$\omega'_0 = \left.\frac{d\omega}{dk}\right|_{k=k_0}~,$$

which explains the group velocity formula.

Other expressions
For light, the refractive index $n$, vacuum wavelength $λ_{0}$, and wavelength in the medium  $λ$, are related by
 * $$\lambda_0 = \frac{2\pi c}{\omega}, \;\; \lambda = \frac{2\pi}{k} = \frac{2\pi v_{\rm p}}{\omega}, \;\; n = \frac{c}{v_{\rm p}} = \frac{\lambda_0}{\lambda},$$

with $v_{p} = ω/k$ the phase velocity.

The group velocity, therefore, can be calculated by any of the following formulas,
 * $$ \begin{align}

v_{\rm g} &= \frac{c}{n + \omega \frac{\partial n}{\partial \omega}} = \frac{c}{n - \lambda_0 \frac{\partial n}{\partial \lambda_0}}\\ &= v_{\rm p} \left(1 + \frac{\lambda}{n} \frac{\partial n}{\partial \lambda}\right) = v_{\rm p} - \lambda \frac{\partial v_{\rm p}}{\partial \lambda} = v_{\rm p} + k \frac{\partial v_{\rm p}}{\partial k}. \end{align}$$

Dispersion
Part of the previous derivation is the Taylor series approximation that:
 * $$\omega(k) \approx \omega_0 + (k - k_0)\omega'_0(k_0)$$

If the wavepacket has a relatively large frequency spread, or if the dispersion $v_{g} = 1⁄2v_{p}$ has sharp variations (such as due to a resonance), or if the packet travels over very long distances, this assumption is not valid, and higher-order terms in the Taylor expansion become important.

As a result, the envelope of the wave packet not only moves, but also distorts, in a manner that can be described by the material's group velocity dispersion. Loosely speaking, different frequency-components of the wavepacket travel at different speeds, with the faster components moving towards the front of the wavepacket and the slower moving towards the back. Eventually, the wave packet gets stretched out. This is an important effect in the propagation of signals through optical fibers and in the design of high-power, short-pulse lasers.

In three dimensions
For waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way: where $$\vec{\nabla}_{\mathbf{k}} \, \omega$$ means the gradient of the angular frequency $ω$ as a function of the wave vector $$\mathbf{k}$$, and $$\hat{\mathbf{k}}$$ is the unit vector in direction k.
 * One dimension: $$v_{\rm p} = \omega/k, \quad v_{\rm g} = \frac{\partial \omega}{\partial k}, \,$$
 * Three dimensions: $$(v_{\rm p})_i = \frac{\omega}{{k}_i}, \quad \mathbf{v}_{\rm g} = \vec{\nabla}_{\mathbf{k}} \, \omega \,$$

If the waves are propagating through an anisotropic (i.e., not rotationally symmetric) medium, for example a crystal, then the phase velocity vector and group velocity vector may point in different directions.

In lossy or gainful media
The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform. However, if the wave is travelling through an absorptive or gainful medium, this does not always hold. In these cases the group velocity may not be a well-defined quantity, or may not be a meaningful quantity.

In his text "Wave Propagation in Periodic Structures", Brillouin argued that in a lossy medium the group velocity ceases to have a clear physical meaning. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon. Another example is mechanical waves in the solar photosphere: The waves are damped (by radiative heat flow from the peaks to the troughs), and related to that, the energy velocity is often substantially lower than the waves' group velocity.

Despite this ambiguity, a common way to extend the concept of group velocity to complex media is to consider spatially damped plane wave solutions inside the medium, which are characterized by a complex-valued wavevector. Then, the imaginary part of the wavevector is arbitrarily discarded and the usual formula for group velocity is applied to the real part of wavevector, i.e.,
 * $$v_{\rm g} = \left(\frac{\partial \left(\operatorname{Re} k\right)}{\partial \omega}\right)^{-1} .$$

Or, equivalently, in terms of the real part of complex refractive index, $ω(k)$, one has
 * $$\frac{c}{v_{\rm g}} = n + \omega \frac{\partial n}{\partial \omega} .$$

It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket. The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real $k$, complex $ω$), or, allow group velocity to be a complex-valued quantity. Different considerations yield distinct velocities, yet all definitions agree for the case of a lossless, gainless medium.

The above generalization of group velocity for complex media can behave strangely, and the example of anomalous dispersion serves as a good illustration. At the edges of a region of anomalous dispersion, $$v_{\rm g}$$ becomes infinite (surpassing even the speed of light in vacuum), and $$v_{\rm g}$$ may easily become negative (its sign opposes Re$k$) inside the band of anomalous dispersion.

Superluminal group velocities
Since the 1980s, various experiments have verified that it is possible for the group velocity (as defined above) of laser light pulses sent through lossy materials, or gainful materials, to significantly exceed the speed of light in vacuum $c$. The peaks of wavepackets were also seen to move faster than $c$.

In all these cases, however, there is no possibility that signals could be carried faster than the speed of light in vacuum, since the high value of $v$$g$ does not help to speed up the true motion of the sharp wavefront that would occur at the start of any real signal. Essentially the seemingly superluminal transmission is an artifact of the narrow band approximation used above to define group velocity and happens because of resonance phenomena in the intervening medium. In a wide band analysis it is seen that the apparently paradoxical speed of propagation of the signal envelope is actually the result of local interference of a wider band of frequencies over many cycles, all of which propagate perfectly causally and at phase velocity. The result is akin to the fact that shadows can travel faster than light, even if the light causing them always propagates at light speed; since the phenomenon being measured is only loosely connected with causality, it does not necessarily respect the rules of causal propagation, even if it under normal circumstances does so and leads to a common intuition.

title: Group velocity


text:
Group action

In mathematics, many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group acts also on triangles by transforming triangles into triangles.

Formally, a group action of a group $C_{3}$ on a set $G$ is a group homomorphism from $S$ to some group (under function composition) of functions from $G$ to itself.

If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it; in particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.

A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group $S$, the group of the invertible matrices of dimension $GL(n, K)$ over a field $n$.

The symmetric group $K$ acts on any set with $Sn$ elements by permuting the elements of the set. Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality.

Left group action
If $G$ is a group with identity element $e$, and $X$ is a set, then a (left) group action $α$ of $G$ on $X$ is a function
 * $$\alpha\colon G \times X \to X,$$

that satisfies the following two axioms:

for all $g$ and $h$ in $G$ and all $x$ in $X$.
 * Identity:
 * $$\alpha(e,x)=x$$
 * Compatibility:
 * $$\alpha(g,\alpha(h,x))=\alpha(gh,x)$$
 * }
 * }

The group $G$ is then said to act on $X$ (from the left). A set $X$ together with an action of $G$ is called a (left) $G$-set.

It can be notationally convenient to curry the action $n$, so that, instead, one has a collection of transformations $α$, with one transformation $α_{g} : X → X$ for each group element $α_{g}$. The identity and compatibility relations then read
 * $$\alpha_e(x) = x$$

and
 * $$\alpha_g(\alpha_h(x)) = (\alpha_g \circ \alpha_h)(x) = \alpha_{gh}(x)$$

with $g ∈ G$ being function composition. The second axiom then states that the function composition is compatible with the group multiplication; they form a commutative diagram. This axiom can be shortened even further, and written as $∘$.

With the above understanding, it is very common to avoid writing $α_{g} ∘ α_{h} = α_{gh}$ entirely, and to replace it with either a dot, or with nothing at all. Thus, $α$ can be shortened to $α(g, x)$ or $g⋅x$, especially when the action is clear from context. The axioms are then
 * $$e{\cdot}x = x$$
 * $$g{\cdot}(h{\cdot}x) = (gh){\cdot}x$$

From these two axioms, it follows that for any fixed $g$ in $G$, the function from $X$ to itself which maps $x$ to $gx$ is a bijection, with inverse bijection the corresponding map for $g⋅x$. Therefore, one may equivalently define a group action of $G$ on $X$ as a group homomorphism from $G$ into the symmetric group $g^{−1}$ of all bijections from $X$ to itself.

Right group action
Likewise, a right group action of $G$ on $X$ is a function
 * $$\alpha\colon X \times G \to X,$$

that satisfies the analogous axioms:

|Identity: |$$\alpha(x,e)=x$$ |Compatibility: |$$\alpha(\alpha(x,g),h)=\alpha(x,gh)$$ (with $Sym(X)$ often shortened to $α(x, g)$ or $xg$ when the action being considered is clear from context)
 * }

|Identity: |$$x{\cdot}e = x$$ |Compatibility: |$$(x{\cdot}g){\cdot}h = x{\cdot}(gh)$$
 * }

for all $g$ and $h$ in $G$ and all $x$ in $X$.

The difference between left and right actions is in the order in which a product $x⋅g$ acts on $x$. For a left action, $h$ acts first, followed by $g$ second. For a right action, $g$ acts first, followed by $h$ second. Because of the formula $gh$, a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a group $G$ on $X$ can be considered as a left action of its opposite group $(gh)^{−1} = h^{−1}g^{−1}$ on $X$.

Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group induces both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.

Notable properties of actions
Let $G^{op}$ be a group acting on a set $G$. The action is called ' or ' if $X$ for all $g⋅x = x$ implies that $x ∈ X$. Equivalently, the homomorphism from $g = e_{G}$ to the group of bijections of $G$ corresponding to the action is injective.

The action is called  (or semiregular or fixed-point free) if the statement that $X$ for some $g⋅x = x$ already implies that $x ∈ X$. In other words, no non-trivial element of $g = e_{G}$ fixes a point of $G$. This is a much stronger property than faithfulness.

For example, the action of any group on itself by left multiplication is free. This observation implies Cayley's theorem that any group can be embedded in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group $X$ (of cardinality $(Z / 2Z)^{n}$) acts faithfully on a set of size $2^{n}$. This is not always the case, for example the cyclic group $2n$ cannot act faithfully on a set of size less than $Z / 2^{n}Z$.

In general the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric group $2^{n}$, the icosahedral group $S_{5}$ and the cyclic group $A_{5} × Z / 2Z$. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.

Transitivity properties
The action of $Z / 120Z$ on $G$ is called  if for any two points $X$ there exists a $x, y ∈ X$ so that $g ∈ G$.

The action is ' (or sharply transitive, or ') if it is both transitive and free. This means that given $g ⋅ x = y$ the element $x, y ∈ X$ in the definition of transitivity is unique. If $g$ is acted upon simply transitively by a group $X$ then it is called a principal homogeneous space for $G$ or a $G$-torsor.

For an integer $G$, the action is if $n ≥ 1$ has at least $X$ elements, and for any pair of $n$-tuples $n$ with pairwise distinct entries (that is $(x_{1}, ..., x_{n}), (y_{1}, ..., y_{n}) ∈ X^{n}$, $x_{i} ≠ x_{j}$ when $y_{i} ≠ y_{j}$) there exists a $i ≠ j$ such that $g ∈ G$ for $g⋅x_{i} = y_{i}$. In other words the action on the subset of $i = 1, ..., n$ of tuples without repeated entries is transitive. For $X^{n}$ this is often called double, respectively triple, transitivity. The class of 2-transitive groups (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally multiply transitive groups is well-studied in finite group theory.

An action is when the action on tuples without repeated entries in $n = 2, 3$ is sharply transitive.

Examples
The action of the symmetric group of $X^{n}$ is transitive, in fact $X$-transitive for any $n$ up to the cardinality of $n$. If $X$ has cardinality $X$, the action of the alternating group is $n$-transitive but not $(n − 2)$-transitive.

The action of the general linear group of a vector space $(n − 1)$ on the set $V$ of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of $V ∖ \{0\}$ is at least 2). The action of the orthogonal group of a Euclidean space is not transitive on nonzero vectors but it is on the unit sphere.

Primitive actions
The action of $v$ on $G$ is called primitive if there is no partition of $X$ preserved by all elements of $X$ apart from the trivial partitions (the partition in a single piece and its dual, the partition into singletons).

Topological properties
Assume that $G$ is a topological space and the action of $X$ is by homeomorphisms.

The action is wandering if every $G$ has a neighbourhood $x ∈ X$ such that there are only finitely many $U$ with $g ∈ G$.

More generally, a point $g⋅U ∩ U ≠ ∅$ is called a point of discontinuity for the action of $x ∈ X$ if there is an open subset $G$ such that there are only finitely many $U ∋ x$ with $g ∈ G$. The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest $g⋅U ∩ U ≠ ∅$-stable open subset $G$ such that the action of $Ω ⊂ X$ on $G$ is wandering. In a dynamical context this is also called a wandering set.

The action is properly discontinuous if for every compact subset $Ω$ there are only finitely many $K ⊂ X$ such that $g ∈ G$. This is strictly stronger than wandering; for instance the action of $g⋅K ∩ K ≠ ∅$ on $Z$ given by $R^{2} ∖ \{(0, 0)\}$ is wandering and free but not properly discontinuous.

The action by deck transformations of the fundamental group of a locally simply connected space on an covering space is wandering and free. Such actions can be characterized by the following property: every $n⋅(x, y) = (2^{n}x, 2^{−n}y)$ has a neighbourhood $x ∈ X$ such that $U$ for every $g⋅U ∩ U = ∅$. Actions with this property are sometimes called freely discontinuous, and the largest subset on which the action is freely discontinuous is then called the free regular set.

An action of a group $g ∈ G ∖ \{e_{G}\}$ on a locally compact space $G$ is called cocompact if there exists a compact subset $X$ such that $A ⊂ X$. For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space $X = G ⋅ A$.

Actions of topological groups
Now assume $G \ X$ is a topological group and $G$ a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map $X$ is continuous for the product topology.

The action is said to be  if the map $G × X → X$ defined by $G × X → X × X$ is proper. This means that given compact sets $(g, x) ↦ (x, g⋅x)$ the set of $K, K′$ such that $g ∈ G$ is compact. In particular, this is equivalent to proper discontinuity $g⋅K ∩ K′ ≠ ∅$ is a discrete group.

It is said to be locally free if there exists a neighbourhood $G$ of $U$ such that $e_{G}$ for all $g⋅x ≠ x$ and $x ∈ X$.

The action is said to be strongly continuous if the orbital map $g ∈ U ∖ \{e_{G}\}$ is continuous for every $g ↦ g⋅x$. Contrary to what the name suggests, this is a weaker property than continuity of the action.

If $x ∈ X$ is a Lie group and $G$ a differentiable manifold, then the subspace of smooth points for the action is the set of points $X$ such that the map $x ∈ X$ is smooth. There is a well-developed theory of Lie group actions, i.e. action which are smooth on the whole space.

Linear actions
If $g ↦ g⋅x$ acts by linear transformations on a module over a commutative ring, the action is said to be irreducible if there are no proper nonzero $g$-invariant submodules. It is said to be  semisimple if it decomposes as a direct sum of irreducible actions.

Orbits and stabilizers

Consider a group $g$ acting on a set $I$. The  of an element $T$ in $I / T$ is the set of elements in $gT$ to which $g$ can be moved by the elements of $G$. The orbit of $X$ is denoted by $x$: $$G{\cdot}x = \{ g{\cdot}x : g \in G \}.$$

The defining properties of a group guarantee that the set of orbits of (points $X$ in) $X$ under the action of $x$ form a partition of $G$. The associated equivalence relation is defined by saying $x$ if and only if there exists a $G⋅x$ in $x$ with $X$. The orbits are then the equivalence classes under this relation; two elements $G$ and $X$ are equivalent if and only if their orbits are the same, that is, $x ~ y$.

The group action is transitive if and only if it has exactly one orbit, that is, if there exists $g$ in $G$ with $g⋅x = y$. This is the case if and only if $x$ for $y$ in $G⋅x = G⋅y$ (given that $x$ is non-empty).

The set of all orbits of $X$ under the action of $G⋅x = X$ is written as $G⋅x = X$ (or, less frequently, as $x$), and is called the  of the action. In geometric situations it may be called the ', while in algebraic situations it may be called the space of ', and written $X$, by contrast with the invariants (fixed points), denoted $X$: the coinvariants are a while the invariants are a. The coinvariant terminology and notation are used particularly in group cohomology and group homology, which use the same superscript/subscript convention.

Invariant subsets
If $X$ is a subset of $G$, then $X / G$ denotes the set $G \ X$. The subset $X_{G}$ is said to be invariant under $X^{G}$ if $Y$ (which is equivalent $X$). In that case, $G⋅Y$ also operates on $\{g⋅y : g ∈ G and y ∈ Y\}$ by restricting the action to $Y$. The subset $G$ is called fixed under $G⋅Y = Y$ if $G⋅Y ⊆ Y$ for all $G$ in $Y$ and all $Y$ in $Y$. Every subset that is fixed under $G$ is also invariant under $g⋅y = y$, but not conversely.

Every orbit is an invariant subset of $g$ on which $G$ acts transitively. Conversely, any invariant subset of $y$ is a union of orbits. The action of $Y$ on $G$ is transitive if and only if all elements are equivalent, meaning that there is only one orbit.

A $G$-invariant element of $X$ is $G$ such that $X$ for all $G$. The set of all such $X$ is denoted $G$ and called the $X$-invariants of $x ∈ X$. When $g⋅x = x$ is a $g ∈ G$-module, $x$ is the zeroth cohomology group of $X^{G}$ with coefficients in $G$, and the higher cohomology groups are the derived functors of the functor of $X$-invariants.

Fixed points and stabilizer subgroups
Given $X$ in $G$ and $X^{G}$ in $G$ with $X$, it is said that "$G$ is a fixed point of $g$" or that "$G$ fixes $x$". For every $X$ in $g⋅x = x$, the  of $x$ with respect to $g$ (also called the isotropy group or little group ) is the set of all elements in $g$ that fix $x$: $$G_x = \{g \in G : g{\cdot}x = x\}.$$ This is a subgroup of $x$, though typically not a normal one. The action of $X$ on $G$ is free if and only if all stabilizers are trivial. The kernel $x$ of the homomorphism with the symmetric group, $G$, is given by the intersection of the stabilizers $x$ for all $G$ in $G$. If $X$ is trivial, the action is said to be faithful (or effective).

Let $N$ and $G → Sym(X)$ be two elements in $G_{x}$, and let $x$ be a group element such that $X$. Then the two stabilizer groups $N$ and $x$ are related by $y$. Proof: by definition, $X$ if and only if $g$. Applying $y = g⋅x$ to both sides of this equality yields $G_{x}$; that is, $G_{y}$. An opposite inclusion follows similarly by taking $G_{y} = gG_{x}g^{−1}$ and $h ∈ G_{y}$.

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, we can associate a conjugacy class of a subgroup of $h⋅(g⋅x) = g⋅x$ (that is, the set of all conjugates of the subgroup). Let $g^{−1}$ denote the conjugacy class of $(g^{−1}hg)⋅x = x$. Then the orbit $g^{−1}hg ∈ G_{x}$ has type $h ∈ G_{x}$ if the stabilizer $x = g^{−1}⋅y$ of some/any $G$ in $(H)$ belongs to $H$. A maximal orbit type is often called a principal orbit type.

and Burnside's lemma
Orbits and stabilizers are closely related. For a fixed $O$ in $(H)$, consider the map $G_{x}$ given by $x$. By definition the image $O$ of this map is the orbit $(H)$. The condition for two elements to have the same image is $$f(g)=f(h) \iff g{\cdot}x = h{\cdot}x \iff g^{-1}h{\cdot}x = x \iff g^{-1}h \in G_x \iff h \in gG_x.$$ In other words, $x$ if and only if $X$ and $f : G → X$ lie in the same coset for the stabilizer subgroup $g ↦ g⋅x$. Thus, the fiber $f(G)$ of $G⋅x$ over any $f(g) = f(h)$ in $g$ is contained in such a coset, and every such coset also occurs as a fiber. Therefore $h$ induces a between the set $G_{x}$ of cosets for the stabilizer subgroup and the orbit $f(\{y\})$, which sends $f$. This result is known as the orbit-stabilizer theorem.

If $y$ is finite then the orbit-stabilizer theorem, together with Lagrange's theorem, gives $$|G \cdot x| = [G\,:\,G_x] = |G| / |G_x|,$$ in other words the length of the orbit of $G⋅x$ times the order of its stabilizer is the order of the group. In particular that implies that the orbit length is a divisor of the group order.

 * Example: Let $f$ be a group of prime order $G / G_{x}$ acting on a set $G⋅x$ with $gG_{x} ↦ g⋅x$ elements. Since each orbit has either $G$ or $x$ elements, there are at most $G$ orbits of length $p$ which are $X$-invariant elements.

This result is especially useful since it can be employed for counting arguments (typically in situations where $k$ is finite as well).

 * Example: We can use the orbit-stabilizer theorem to count the automorphisms of a graph. Consider the cubical graph as pictured, and let $1$ denote its automorphism group. Then $p$ acts on the set of vertices $k mod p$, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, $1$. Applying the theorem now to the stabilizer $G$, we can obtain $X$. Any element of $G$ that fixes 1 must send 2 to either 2, 4, or 5. As an example of such automorphisms consider the rotation around the diagonal axis through 1 and 7 by $G$, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, $\{1, 2, ..., 8\}$. Applying the theorem a third time gives $|G| = |G ⋅ 1| |G_{1}| = 8 |G_{1}|$. Any element of $G_{1}$ that fixes 1 and 2 must send 3 to either 3 or 6. Reflecting the cube at the plane through 1, 2, 7 and 8 is such an automorphism sending 3 to 6, thus $|G_{1}| = |(G_{1}) ⋅ 2| |(G_{1})_{2}|$. One also sees that $G$ consists only of the identity automorphism, as any element of $2π/3$ fixing 1, 2 and 3 must also fix all other vertices, since they are determined by their adjacency to 1, 2 and 3. Combining the preceding calculations, we can now obtain $|(G_{1}) ⋅ 2| = 3$.

A result closely related to the orbit-stabilizer theorem is Burnside's lemma: $$|X/G|=\frac{1}{|G|}\sum_{g\in G} |X^g|,$$ where $|(G_{1})_{2}| = |((G_{1})_{2}) ⋅ 3| |((G_{1})_{2})_{3}|$ is the set of points fixed by $G$. This result is mainly of use when $|((G_{1})_{2}) ⋅ 3| = 2$ and $((G_{1})_{2})_{3}$ are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.

Fixing a group $G$, the set of formal differences of finite $|G| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48$-sets forms a ring called the Burnside ring of $X^{g}$, where addition corresponds to disjoint union, and multiplication to Cartesian product.

Examples

 * The  action of any group $g$ on any set $G$ is defined by $X$ for all $G$ in $G$ and all $G$ in $G$; that is, every group element induces the identity permutation on $X$.
 * In every group $g⋅x = x$, left multiplication is an action of $g$ on $G$: $x$ for all $X$, $X$ in $G$. This action is free and transitive (regular), and forms the basis of a rapid proof of Cayley's theorem – that every group is isomorphic to a subgroup of the symmetric group of permutations of the set $G$.
 * In every group $G$ with subgroup $g⋅x = gx$, left multiplication is an action of $g$ on the set of cosets $x$: $G$ for all $G$, $G$ in $H$. In particular if $G$ contains no nontrivial normal subgroups of $G / H$ this induces an isomorphism from $g⋅aH = gaH$ to a subgroup of the permutation group of degree $g$.
 * In every group $a$, conjugation is an action of $G$ on $H$: $G$. An exponential notation is commonly used for the right-action variant: $G$; it satisfies ($[G : H]$.
 * In every group $G$ with subgroup $G$, conjugation is an action of $G$ on conjugates of $g⋅x = gxg^{−1}$: $x^{g} = g^{−1}xg$ for all $x^{g})^{h} = x^{gh}$ in $G$ and $H$ conjugates of $G$.
 * An action of $H$ on a set $g⋅K = gKg^{−1}$ uniquely determines and is determined by an automorphism of $g$, given by the action of 1. Similarly, an action of $G$ on $K$ is equivalent to the data of an involution of $H$.
 * The symmetric group $Z$ and its subgroups act on the set $X$ by permuting its elements
 * The symmetry group of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.
 * The symmetry group of any geometrical object acts on the set of points of that object.
 * For a coordinate space $X$ over a field $Z / 2Z$ with group of units $X$, the mapping $X$ given by $S_{n}$ is a group action called scalar multiplication.
 * The automorphism group of a vector space (or graph, or group, or ring ...) acts on the vector space (or set of vertices of the graph, or group, or ring ...).
 * The general linear group $\{1, ..., n\}$ and its subgroups, particularly its Lie subgroups (including the special linear group $V$, orthogonal group $F$, special orthogonal group $F*$, and symplectic group $F* × V → V$) are Lie groups that act on the vector space $a × (x_{1}, x_{2}, ..., x_{n}) ↦ (ax_{1}, ax_{2}, ..., ax_{n})$. The group operations are given by multiplying the matrices from the groups with the vectors from $GL(n, K)$.
 * The general linear group $SL(n, K)$ acts on $O(n, K)$ by natural matrix action. The orbits of its action are classified by the greatest common divisor of coordinates of the vector in $SO(n, K)$.
 * The affine group acts transitively on the points of an affine space, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, regular) action on these points; indeed this can be used to give a definition of an affine space.
 * The projective linear group $Sp(n, K)$ and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the projective space $K^{n}$. This is a quotient of the action of the general linear group on projective space. Particularly notable is $K^{n}$, the symmetries of the projective line, which is sharply 3-transitive, preserving the cross ratio; the Möbius group $GL(n, Z)$ is of particular interest.
 * The isometries of the plane act on the set of 2D images and patterns, such as wallpaper patterns. The definition can be made more precise by specifying what is meant by image or pattern, for example, a function of position with values in a set of colors. Isometries are in fact one example of affine group (action).
 * The sets acted on by a group $Z^{n}$ comprise the category of $Z^{n}$-sets in which the objects are $PGL(n + 1, K)$-sets and the morphisms are $P^{n}(K)$-set homomorphisms: functions $PGL(2, K)$ such that $PGL(2, C)$ for every $G$ in $G$.
 * The Galois group of a field extension $G$ acts on the field $G$ but has only a trivial action on elements of the subfield $f : X → Y$. Subgroups of $g⋅(f(x)) = f(g⋅x)$ correspond to subfields of $g$ that contain $G$, that is, intermediate field extensions between $L / K$ and $L$.
 * The additive group of the real numbers $K$ acts on the phase space of "well-behaved" systems in classical mechanics (and in more general dynamical systems) by time translation: if $Gal(L / K)$ is in $L$ and $K$ is in the phase space, then $L$ describes a state of the system, and $K$ is defined to be the state of the system $(R, +)$ seconds later if $t$ is positive or $R$ seconds ago if $x$ is negative.
 * The additive group of the real numbers $x$ acts on the set of real functions of a real variable in various ways, with $t + x$ equal to, for example, $t$, $t$, $−t$, $t$, $(R, +)$, or $(t⋅f)(x)$, but not $f(x + t)$.
 * Given a group action of $f(x) + t$ on $f(xe^{t})$, we can define an induced action of $f(x)e^{t}$ on the power set of $f(x + t)e^{t}$, by setting $f(xe^{t}) + t$ for every subset $f(xe^{t} + t)$ of $G$ and every $X$ in $G$. This is useful, for instance, in studying the action of the large Mathieu group on a 24-set and in studying symmetry in certain models of finite geometries.
 * The quaternions with norm 1 (the versors), as a multiplicative group, act on $X$: for any such quaternion $g⋅U = {g⋅u : u ∈ U}$, the mapping $U$ is a counterclockwise rotation through an angle $X$ about an axis given by a unit vector $g$; $G$ is the same rotation; see quaternions and spatial rotation. This is not a faithful action because the quaternion $R^{3}$ leaves all points where they were, as does the quaternion $z = cos α/2 + v sin α/2$.
 * Given left $f(x) = zxz^{*}$-sets $α$, $v$, there is a left $z$-set $−1$ whose elements are $1$-equivariant maps $G$, and with left $X$-action given by $Y$ (where "$G$" indicates right multiplication by $Y$). This $G$-set has the property that its fixed points correspond to equivariant maps $α : X × G → Y$; more generally, it is an exponential object in the category of $G$-sets.

Group actions and groupoids
The notion of group action can be encoded by the action groupoid $g⋅α = α ∘ (id_{X} × –g)$ associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.

Morphisms and isomorphisms between G-sets
If $–g$ and $g$ are two $G$-sets, a morphism from $X → Y$ to $G$ is a function $G′ = G ⋉ X$ such that $X$ for all $Y$ in $G$ and all $X$ in $Y$. Morphisms of $f : X → Y$-sets are also called equivariant maps or $f(g⋅x) = g⋅f(x)$-maps.

The composition of two morphisms is again a morphism. If a morphism $g$ is bijective, then its inverse is also a morphism. In this case $G$ is called an isomorphism, and the two $x$-sets $X$ and $G$ are called isomorphic; for all practical purposes, isomorphic $G$-sets are indistinguishable.

Some example isomorphisms:
 * Every regular $f$ action is isomorphic to the action of $f$ on $G$ given by left multiplication.
 * Every free $X$ action is isomorphic to $Y$, where $G$ is some set and $G$ acts on $G$ by left multiplication on the first coordinate. ($G$ can be taken to be the set of orbits $G$.)
 * Every transitive $G × S$ action is isomorphic to left multiplication by $S$ on the set of left cosets of some subgroup $G$ of $G × S$. ($S$ can be taken to be the stabilizer group of any element of the original $X / G$-set.)

With this notion of morphism, the collection of all $G$-sets forms a category; this category is a Grothendieck topos (in fact, assuming a classical metalogic, this topos will even be Boolean).

Variants and generalizations
We can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See semigroup action.

Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object $G$ of some category, and then define an action on $H$ as a monoid homomorphism into the monoid of endomorphisms of $G$. If $H$ has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion.

We can view a group $G$ as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant) functor from $G$ to the category of sets, and a group representation is a functor from $X$ to the category of vector spaces. A morphism between $X$-sets is then a natural transformation between the group action functors. In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.

In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds, regular actions of algebraic groups on algebraic varieties, and actions of group schemes on schemes. All of these are examples of group objects acting on objects of their respective category.

title: Group action