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Raytheon Missiles & Defense

Raytheon Missiles & Defense (RMD) was one of four business segments of RTX Corporation. Headquartered in Tucson, Arizona, its president was Wes Kremer. The business produced a broad portfolio of advanced technologies, including air and missile defense systems, precision weapons, radars, and command and control systems. Raytheon Intelligence & Space was merged with Raytheon Missiles & Defense in July 2023 to form the Raytheon business segment.

History
The business was a combination of two Raytheon Company legacy businesses, Raytheon Integrated Defense Systems (IDS) and Raytheon Missile Systems (RMS), which operated a plant formerly owned by the Hughes Aircraft Company.

Raytheon had been criticized for selling arms to Saudi Arabia that were used in the Yemen Civil War. These sales were blocked by the Obama administration in 2016 due to humanitarian concerns, a decision that was reversed by the Trump administration six months later.

Products
Key Raytheon Missiles & Defense capabilities combined key IDS and RMS capabilities. Key IDS capabilities include: Key RMS capabilities include:
 * Ground-based and sea-based radars for air and missile defense
 * Navy radar and sonar
 * Torpedoes and naval mine countermeasures
 * Missiles and precision-guided munitions
 * Drones and air-launched decoys
 * Counter-drone and non-lethal directed energy weapons
 * Ground vehicle sensors and weapons
 * Ground based operational surveillance system

The division's products included:
 * Active Denial System non-lethal millimeter wave weapon
 * AGM-65 Maverick air-to-surface missile
 * AGM-88 HARM air-to-surface missile
 * AGM-129 ACM air-to-surface missile
 * AGM-154 Joint Standoff Weapon air-to-surface glide bomb
 * AGM-176 Griffin air-to-surface missile
 * AIM-7 Sparrow air-to-air missile
 * RIM-7 Sea Sparrow naval surface-to-air missile
 * AIM-9 Sidewinder air-to-air missile
 * AIM-54 Phoenix air-to-air missile
 * AIM-120 AMRAAM air-to-air missile
 * AN/SPY-6 Air and Missile Defense Radar (AMDR) for Navy ships
 * AN/TPY-2 radar for the THAAD missile defense system
 * AN/AQS20C Mine hunting sonar suite
 * AN/ASQ-235 Airborne Mine Neutralization System (AMNS)
 * BGM-71 TOW anti-tank missile
 * BGM-109 Tomahawk cruise missile
 * Coyote unmanned aerial system
 * David's Sling Medium to long range surface-to-air/anti-ballistic missile (partner with prime contractor Rafael Advanced Defense Systems)
 * Extended Range Guided Munition
 * Exoatmospheric Kill Vehicle anti-ICBM system
 * FGM-148 Javelin anti-tank missile
 * FIM-92 Stinger person-portable air defense system surface-to-air missile
 * M982 Excalibur guided artillery round
 * MIM-23 Hawk surface-to-air missile
 * MIM-104 Patriot surface-to-air missile
 * Paveway laser-guided bomb
 * Phalanx CIWS naval anti-missile defense system
 * RIM-116 Rolling Airframe Missile naval surface-to-air missile
 * RIM-162 Evolved Sea Sparrow Missile naval surface-to-air missile
 * Standard Missile family of naval missiles
 * RIM-66 Standard
 * RIM-67 Standard
 * RIM-161 Standard Missile 3
 * RIM-174 Standard ERAM
 * SAM-N-2 Lark

title: Raytheon Missiles & Defense


text:
TDD

TDD may refer to:

Science and technology

 * Telecommunications device for the deaf, a device for text communication along a telephone line
 * Test-driven development, a type of software development methodology
 * Time-division duplex, the application of time-division multiplexing to separate outward and return signals
 * LTE-TDD, a 4G telecommunications technology and standard
 * Transdermal drug delivery, a method of delivering drugs through the skin and into the bloodstream

Other uses

 * Transportation development district, a special-purpose district created in some US states
 * Teniente Jorge Henrich Arauz Airport (IATA airport code), Trinidad, Bolivia
 * Tai Nüa language (ISO639-3 code: tdd)

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

title: TDD


text:
Bobbejaan Schoepen

Bobbejaan Schoepen (a pseudonym of Modest Schoepen; 16 May 1925 – 17 May 2010) was a Flemish pioneer in Belgian pop music, vaudeville, and European country music. Schoepen was a versatile entertainer, entrepreneur, singer-songwriter, guitarist, comedian, actor, and professional whistler, as well as the founder and former director of the amusement park Bobbejaanland. His musical career flourished from 1948 until the first half of the 1970s. He sold more than five million copies from his repertoire of 482 songs, which extended from Twang, cabaret, instrumental film music, chansons, country, to folk and vocal music. Born in Boom, Antwerp, Flanders, Belgium, he worked his way up from a working-class environment to become one of the 200 richest people in Belgium.

Schoepen married Dutch former opera singer and photographer's model Josephina (Josée) Jongen on 18 May 1961. They have five children: Robert ("Bob Jr.", 1962), Myriam (1963), Jacky (1964), Peggy (1968), and Tom (1970). His son Tom became his manager in Belgium.

Early musical period
Modest Schoepen grew up in a smithy in Boom, Antwerp. His career started in the late 1930s when he and his sister Liesje performed vaudeville shows in the surrounding villages, going around with the hat collecting money afterwards. He had his first audition for radio, in 1944 in Brussels. In 1943 he undertook classical guitar instruction with guitarist Frans De Groodt (1892–1990).

Musical and political debut
That same year he had a memorable debut performance in the Ancienne Belgique of Brussels. In front of a full house he sang the South-African song, "Mama, ek wil 'n man hé". Nee mamma, née, 'n Duitseman, die wil ek nie. Want Schweinefleisch dit lus ek nie" ("Mommy, I want a husband. No mommy no, I don't want a German 'cause I don't like pig meat.") The song was perceived as being anti-German, provoking a few Nazis who were present at the show to take him away. The South African song is making reference to a 'German man'. The Ancienne Belgique was closed for three weeks. Shortly thereafter he was forced to go work in Germany. As an alternative he chose to sing for the Flemish workers doing compulsory labor. For this he was locked up for three months in the Dossin barracks in Mechelen from October 1944, without a hearing or a trial.

Two Boys and Two Guitars
In 1945 he formed a duo with Kees Brug, a young man from his own village, with the name of "Two Boys and Two Guitars". They performed impersonations, poetry, South-African songs, and country music from Calais to Amsterdam, all with plenty of room for improvisation and adventure. The name "Bobbejaan" comes from the South-African song, "Bobbejaan klim die berg" ("Baboon Climbs the Mountain"). Schoepen took it as his artist name in 1945 or 1946.

Performing to the allied forces
In 1947 he came into contact with Jacques Kluger. Kluger asked Schoepen to entertain the American and Canadian troops during the Nuremberg Trials, and in Frankfurt and Berlin. Kluger was pleased to receive an unexpected, flattering letter from Major Mearker, and contracted Schoepen to go on tour in Germany for several months. In Berlin, which was still partly in ruins, his floor shows were also attended by the American general and military governor Lucius D. Clay, who asked him for two additional performances. These tours would further stimulate his country music tendencies.

First hit record and Indonesian tour
In between these shows, Schoepen also gave occasional performances in his own country. While he initially did not wish to sing in Dutch, Kluger convinced him to record 'a Flemish record'. Schoepen's first recordings followed, and in 1948 "De Jodelende Fluiter" ("The Yodeling Whistler") became Schoepen's first hit. His breakthrough in the Netherlands also happened that year. Among his many performances, he was frequently asked to guest-star in the Netherlands.

Bobbejaan soon became the right man to work for the benefit of the Dutch. In 1949 he went on tour for the Dutch troops in Indonesia, performing 127 shows over the course of three months. He was decorated for courage and self-sacrifice by the Dutch government because he also performed for troops near the front line. Five days after returning home he began a 220-day tour through Belgium, playing nostalgic songs, such as the tried and true "De lichtjes van de Schelde" ("The Lights of the Scheldt") (1952), which is still an evergreen in Belgium today, Bobbejaan Schoepen quickly became one of the most popular artists in Flanders.

International tours and success
Schoepen toured in at least twenty different countries, together with artists such as Josephine Baker, Caterina Valente, Gilbert Bécaud, and Toots Thielemans (who was a guitarist in his band in 1951). He is one of the first Europeans (excluding the British) to have appeared at the Grand Ole Opry in Nashville, one of the most important centers of country music in the United States. In 1953 he played there three times with Roy Acuff (1903–1992). There was one performance with the country singer Red Foley (1910–1968) in Springfield, Missouri. The American country singer, Tex Williams, a Western swing performer, would later release a cover of Schoepen's "Fire and Blisters" in the US (1974).

1954 tours
In 1954 there followed a three-month European tour through Germany, Iceland, and Denmark, which concluded with a few months of performances in the Folies Bergère in Brussels. Syd Fox was Schoepen's manager in Iceland and Denmark. When Jacques Brel performed the opening act in the Ancienne Belgique in Brussels in January 1955, Schoepen already had the status of an international vedette in his own country. He was chosen as the best Flemish singer by the broadcasting company NIR (former VRT), for which he received the "Grand Prize for Flemish Gramophone Recording."

That autumn he toured with his show for a month through Germany then a month through the Congo.

1957 tours
In 1957 he again went to New York, where he was asked to make a guest-appearance on the famous Ed Sullivan Show. He recorded albums at RCA Records with the producer Steve Sholes. Sholes offered him a contract to promote his record, by visiting radio stations in the US for three months, under the name "Bobby John". This tour was intended to promote his recent releases, and in the meantime he was to produce new songs. But Schoepen, who had already been on tour almost constantly for ten years, and had contractual obligations in Europe, was increasingly looking for a place to settle down (this would become Bobbejaanland). He decided no longer to pursue his success in the US.

That same month he became the last-minute Belgian representative at the second Eurovision Song Contest 1957. He was rushed by his manager Jacques Kluger from the US to Germany in order to take part in the event. This performance was memorable for having featured a whistling solo. Schoepen is also rumoured not to have known which song he was to perform at the Contest until he arrived, only rehearsing his entry a few days before performance. The song that was finally chosen was the lightly poetic "Straatdeuntje" ('Street Tune'). Belgium tied for eighth place with the Swiss.

1958 tours
A year later in 1958, Kluger secured Schoepen a place in the "Royal Variety Show" in England, a yearly gala for the Queen Mother (Elizabeth Bowes-Lyon). After the show, his local manager Jack Heath let him hear the first golden Australian hit in the US, "A Pub with No Beer" by Slim Dusty, and Schoepen decided to make a Dutch, German, and English cover of it. In 1960, "Ich steh an der Bar und ich habe kein Geld" stayed on the hit lists for thirty weeks in Germany; it also became a number-one hit in Austria. The Flemish version, "Café zonder bier" dates from 1959, and that year it reached @3 in the hit parade. It later became a golden oldie.

Other hits and tours
His German versions of "Een hutje op de Heide" ("Little cottage on the Heath") and "Kili Watch" (originally by The Cousins) also did very well (among other reasons because of the German film, Davon träumen alle Mädchen, 1961). Schoepen toured frequently in Germany and Austria with Caterina Valente (1955) and Dalida (1961), among others. This earned him many new, interesting contracts. In 1961 he became one of the top musical acts at the Berlin Film Festival, where he brought down the house at the Deutschlandhalle with his crazy whistling acts.

During the 1960s, Camillo Felgen, Heino, and James Last, among others, would catapult his evergreen "Ik heb eerbied voor jouw grijze haren" ("I Respect Your Grey Hair") into a huge European hit, which has sold more than three million copies. In 1961, Caterina Valente also released a version of "In de schaduw van de mijn" ("In the Shadow of the [Coal] Mine") in Italy, under the title "Amice Miei", and in 1965 Richard Anthony sang a French and Spanish version "Ik heb me dikwijls afgevraagd" (Je me suis souvent demandé) into the international charts: the song reached #1 in France, April 3, 1965. In 1965 it earned Schoepen, who wrote the music, an artistic 'Diploma of the Croix d'Honneur' in Paris.

In 1967, ZDF-Germany decided to produce a musical television film in which he plays a series of his hits. The film was partially recorded in the Barrandov Studios in Prague, in Kempen and in Bobbejaanland, which at the time was still without attractions.

During the sixties and seventies, he was a regular visitor to the United States, where he took up with actor Roy Rogers, Nudie Cohn (the fashion designer of Elvis, Johnny Cash and others), and Tex Williams, the founding father of western swing. The foursome occasionally performed together in local clubs.

Tours with a circus tent: 1958–1961
In 1958, Schoepen purchased a big circus tent to be able to tour more efficiently in his own country. Doing so freed him from having to deal with music venue owners, who were asking for ever-increasing rental prices and did not always have appropriate space for his program. He took over the two-master circus tent from the Tondeur family, who were finding it difficult to keep their performances profitable. Schoepen was handed the organizational reins and the circus again began to flourish. The day before a show he would promote the show himself at the designated location, sometimes in his own bizarre manner.

An American circus stunt team came to Brussels on the occasion of the world exhibition, Expo 58; it was a private endeavor led by Casey Tibbs. But because of the excessively long distance to the actual Expo events, and the persistent bad weather, they went bankrupt. They could no longer afford to care for their fifty horses and so the team had to return to the US. Tibbs had no choice but to give away some of the equipment, and among other things he sold Zorro's horse—named Midnight, from the Zorro television series—to Bobbejaan Schoepen. He used the horse for many years for stunts in his shows and as an attraction during his cavalcades, but the horse stepped on an exposed electricity cable and died.

In 1959, Schoepen bought a new circus tent. It had room for 900 people and could be expanded to accommodate a capacity of 1,200. These tours came to an end as soon as Bobbejaanland opened its doors in 1961. Using a circus tent for concert tours was unique for its time.

Film
Between 1950 and 1967 he acted in five musical film productions: two Belgian, two German, and one German-Czech. In 1962 he played the leading role in the absurd comedy, At the Drop of a Head (alias De Ordannans), together with leaders of the Flemish film scene: Ann Peterson, Yvonne Lex, Denise Deweerdt, Nand Buyl, and Tony Bell. The Dutch and English versions were recorded on the set at the same time. Schoepen was not pleased with this particular adventure into the world of film: "The takes were chaotic and they fired two different directors. Jef Bruyninckx (alias, De Witte) had to solve everything." In 1999, the Belgian cult-rock band, Dead Man Ray, toured through Belgium and the Netherlands with the film. For Daan Stuyven (Daan) and Rudy Trouvé (ex-dEUS), it is also an ode to the sometimes misunderstood)artistic versatility that characterized Schoepen the artist:
 * A true professional who was able to turn his jazzy country-guitar playing, his deep, angelic voice and his wacky sense of humor into a trademark and later into an amusement park.

Bobbejaanland and caberet
It was never Schoepen's intention to build an amusement park; the current park arose from his musical career. After nearly 15 years of constant touring, he began thinking about a place to settle down. In 1959, he bought a 30 hectare, marshy domain in Lichtaart-Kasterlee, called the Abroek. There he built a theater with nearly 1000 seats and created a 2.2 kilometer beach. This was to become Bobbejaanland, a name coined by his manager, Jacques Kluger. On 31 December 1961, Bobbejaanland was officially opened by Bobbejaan Schoepen and his wife Josée, with whom he developed the park into his life's work. During the high season, Schoepen himself performed from two to five concerts per day. Artists from the Belgian, Dutch, and German cabaret-world performed there as well, though not during Schoepen's own program.

Bobbejaanland evolved from 1975 onwards into a fully fledged amusement park. Schoepen focused his attention solely on the business aspect of the park, slowly pushing his musical career into the background. His performances became a rarity but Bobbejaanland, which he operated with his family, developed into one of the top amusement parks in the Benelux region.

In the mid-80s the shows became more streamlined, and oriented toward a more international audience. The amusement park finally dominated the cabaret, and the entrepreneur the artist. He did continue singing, albeit with more streamlined shows, which inevitably gave way to the routine of the amusement park.
 * At a certain point my performances no longer had anything artistic about them. I received a phone call from my wife five minutes before I had to go on stage: 'Two busses of Germans, one bus of Danes and three busses of Spaniards'. And then I adapted my show to suit them. There was no room for sentiment in Bobbejaanland. It was work, work, and more work in order to keep the business profitable. Sixteen hours a day, seven days a week. We employed four-hundred people who had to be paid every month.

Challenges
Bobbejaan Schoepen's life was not without its difficulties: he was thrown into prison twice during wartime, he lost his virtuoso whistling ability due to a surgical intervention, and in 1986 he underwent a serious heart operation. In 1999 he was diagnosed with intestinal cancer, which raised the issue of selling Bobbejaanland, his life's work. But he continued and, in the winter of 2003, a major investment of nearly 12 million euros was made on two unique rides ("Typhoon" and "Sledge Hammer"). That year, the Flemish consumer magazine Test-Aankoop conducted a comparative survey of 13 European amusement parks. Bobbejaanland was the most highly esteemed park in Belgium on almost every level, and in Europe it shared second place, together with Disneyland and the Parc Astérix.

However, in April 2004, after preparing for more than three years, the family finally decided to sell the park. At that time Bobbejaanland employed 400 people and was taken over by Parques Reunidos, a Spanish-American amusement park group. It remained uncertain until the very last minute whether the founder would put his signature on the deal. Uncertainty over the future of amusement parks was ultimately decisive. The Schoepen family chose the more certain path, which also seemed evident to the founder. After the sale, when the news program Terzake on the Flemish Broadcast VRT asked him if he had chosen to take the big money, Schoepen answered, staring ahead blankly, "What am I going to do with all that money; I can only eat twice a day."

Schoepen was considered to be the artistic attraction of the park; his wife Josée (the oldest of eighteen children) was a leading figure and the commercial backbone; and her sister Louise (Wies) was the solid foundation with respect to accounting and finances.
 * The key to the success of this family business is found in the loyalty within this triumvirate, as well as their work ethic, which was strengthened from the 1990s on by three of their five children.

With the sale of the park, the last family business in the amusement park sector in Belgium disappeared. Schoepen and his wife lived on the domain together, until his death in May 2010.

Comeback
Schoepen recovered from his illness and was again engaged in music. In 2005 he gave four surprise performances at the literary festival Saint-Amour, where he again brought people his well-known "De lichtjes van de Schelde". The song dated from 1952 and since then has been covered many times by popular Flemish and Dutch artists such as Louis Neefs, Hans De Booij, Wannes Van de Velde, and recently by Will Tura and DAAN. In November 2006 at the Radio2 "Hall of Fame", the song was performed by Bobbejaan admirer Daan Stuyven. That month, with the efforts of his son Tom Schoepen, Bobbejaan Records was brought back to life. The record company was established in 1966 but lapsed following the success of Bobbejaanland. The label is now focused on releases centered on Bobbejaan Schoepen.

Award
On 13 February 2007, with much media attention, Schoepen received a Lifetime Achievement Award in the Ancienne Belgique in Brussels for his successful career as a singer, musician, and for his pioneering work in Belgian music history.

Another album
After Bobbejaanland was sold, Schoepen focussed on his music career, resulting in the new album release Bobbejaan (Bobbejaan Records/PIAS) in May 2008, his first album in 35 years. The album was mixed at South Beach Studios in Miami.

The idea for this project originated with music producer Firmin Michiels and the Dead Man Ray-tour in 1999, but the idea was shelved when Schoepen was diagnosed with cancer. In 2005, the idea was revived by Michiels (A&R) and executive producer Tom Schoepen, who would concentrate fully on the production of Bob's voice recordings. Michiels was aware that Bobbejaan Schoepen could not just pick up where he left off and carry his show for a large audience: his top years were long past and the amusement park had inflicted too much damage on the singer. But slowly his credibility returned. The album was recorded with five Belgian musicians in Schoepen's living room; he would have to deal with various health issues throughout the recording sessions. The release in May 2008 immediately received broad attention from the TV-journals and other media.

Final years and death
In July 2008 he was inducted as the first European into the "Whistler's Hall of Fame", by the US International Whistlers Convention.

In December 2009, an official compilation with 76 songs spanning 60 years of musical career was released: "The World of Bobbejaan — Songbook" (Bobbejaan Records).

Schoepen died on 17 May 2010, the day after his 85th birthday, from cardiac arrest.

In other media

 * He had his own comic strip: "De Bobbejaanstory" ("The Bobbejaan Story"), by Jef Broeckx, Jacques Bakker, and Ronnie Van Riet.
 * He also appeared in the comic book "Jommeke in Bobbejaanland", from the series Jommeke by Jef Nys (1978).
 * He also appeared in the comic strip "Urbanus: de pretparkprutsers" ("Urbanus: The Amusement Park Bunglers"), Urbanus and Linthout, published by Uitgeverij Loempia.
 * In 2005, Schoepen was one of the contestants for the title of The Greatest Belgian, but did not make the final list of nominees and ended up 307th among those who fell outside the nominees.

Awards and nominations Bobbejaan Schoepen

 * 1949: Decorated "For courage and self-sacrifice" for his musical support of Dutch soldiers on the front lines in Indonesia, presented by General Baay, commander of Dutch troops in East Java, 1949.
 * 1955: Chosen as best Flemish singer, won the "Grand Prize for Flemish Gramophone Recording" (BRT) in cooperation with Studio Gent. (15 March 1955)
 * 1965: Education Artistique, diplôme de Croix d'Honneur de Chevalier. Presented by the Académie Nationale Artistique Littéraire et Scientifique in Paris (No 5177). (30 June 1965)
 * 1978: Platinum record for 30 years of Flemish Hits, Telstar Holland, 1978.
 * 1986: Companion of the Order of the Crown, presented by the Ministry of the Flemish Community. (9 April 1986)
 * 1992: Bobbejaanland receives the "Brass Ring Award" (1st prize for best advertising brochure) from the IAAPA in Dallas (US).
 * 1993: Sabam Prize-medal – Belgian Artistical Promotion. (19 January 1993)
 * 1995: Sabam Prize-medal – Belgian Artistical Promotion. (26 September 1995)
 * 1995: Knight of the Order of Leopold II, presented by the Ministry of the Flemish Community. (26 September 1995)
 * 2000: Place of honour in the Radio 2 & Sabam Hall of Fame. (November 2000)
 * 2000: "Ik heb eerbied voor jouw grijze haren" (nomination for the Radio2 Hall of Fame, 2000)
 * 2005: "De lichtjes van de Schelde" (nomination for the Radio2 Hall of Fame, November 2005)
 * 2006: "De lichtjes van de Schelde" (Radio 2 & Sabam Hall of Fame Award, November 2006)
 * 2007: "Lifetime Achievement Award", ZAMU 2006. (13 February 2007)
 * 2008: "Inducted into the US Whistlers Hall of Fame" presented by the International Whistlers Convention in Tokyo. (21 July 2008)
 * 2009: Officer in the Order of the Crown, presented by the Ministry of the Flemish Community. (6 July 2009)
 * 2009: First honorary citizen of Boom, on the occasion of the 700th anniversary of this municipality. (2 October 2009)
 * Overall: 25 Golden Records.

Five international greatest hits

 * "Ich hab Ehrfurcht vor schneeweißen Haaren" (1959 and 1960s)
 * "Je me suis souvent demandé" (1965)
 * "Ich steh an der Bar und habe kein Geld" (1959) ("A Pub with No Beer")
 * "De lichtjes van de Schelde" (1952, Belgium)
 * "Ein Hauschen auf der Heide" (1960)/"Kili watch" (1961)

Artist names

 * Belgium and The Netherlands: Bobbejaan Schoepen
 * Germany and Austria: Bobby Jaan, Bobbejaan
 * Denmark and Iceland: Bobby Jaan
 * France: Bobby Jaan, Bobby Jann, Bobbi-Jean
 * US: Bobby John

Filmography

 * Ah! t'Is zo fijn in België te leven (1950, Belgium)
 * Televisite (TV series 1955, Belgium)
 * The Eurovision Song Contest (1957, West Germany)
 * At the Drop of a Head/De Ordonnans/Café zonder bier (1962, Belgium–England)
 * O sole mio (1960 film) (1960, West Germany)
 * Davon träumen alle Mädchen (1961, West Germany)
 * Bobbejaanland (film production ZDF – by Vladimir Sis, 1967, Studio Barrandov Prague)
 * Der Goldene Schuß—TV episode (as Bobbejaan) (Musical, 1969)
 * "Uit met Bobbejaan" (BRT, 1969)
 * "30 jaar Bobbejaan" (BRT, 1978)
 * "Bobbejaan 70" (Vlaamse Radio- en Televisieomroep, 1995)

title: Bobbejaan Schoepen


text:
The Land Girls

The Land Girls is a 1998 film directed by David Leland and starring Catherine McCormack, Rachel Weisz, Anna Friel, Steven Mackintosh and Ann Bell. It is based on the 1995 novel Land Girls by Angela Huth.

The title refers to the real-life British women who were called upon to assist rural Englanders when men left their farms to fight in the First and Second World Wars.

Plot
During both the First and Second World Wars, the Women's Land Army was set up in the United Kingdom to recruit women to work at farms where men had left to go to war. Women in the WLA were nicknamed "land girls".

Set in 1941 in the Dorset countryside, three "land girls" arrive on a remote farm. They are an unlikely trio: hairdresser Prue is vivacious and sexy, Cambridge University graduate Ag is quiet and more reserved, and dreamy Stella is in love with Philip, a dashing Royal Navy officer. Despite the women's differences, they soon become close friends. The film follows their relationships with each other and the men in their lives in the face of war.

Cast

 * Catherine McCormack as Stella
 * Rachel Weisz as Ag (Agapanthus)
 * Anna Friel as Prue (Prudence)
 * Steven Mackintosh as Joe Lawrence
 * Tom Georgeson as Mr Lawrence
 * Maureen O'Brien as Mrs Lawrence
 * Lucy Akhurst as Janet
 * Gerald Down as Ratty, Lawrence Farm Hand
 * Paul Bettany as Philip

Production
Filming locations included the scenic Exmoor National Park, Crowcombe Heathfield station on the West Somerset Railway and Dulverton.

The film cost £6 million.

Reception
The film holds a 61% on Rotten Tomatoes based on 18 reviews.

In a review for The New York Times, Stephen Holden wrote, "The movie is ultimately more passionate about its characters' place in history than about their individual lives, which take fairly predictable if not always happy turns. In the end, all three women blend together as an idealized everywoman lightly sketched on a larger historical canvas."

Holden concluded, "The film's most evocative historical set piece shows a parade and air show at which a newly built Spitfire makes its maiden flight. The movie is unequivocal about the nobility of this moment. The shared sacrifice and hard labor have all been worth it. As the aircraft soars and dips across the rural English landscape, it is the very embodiment of a glorious winged victory soon to come."

Roger Ebert praised the cinematography in particular, stating "What I liked about the movie--what I preferred to the romances and relationships--was its look, its sensual evocation of the British countryside in winter." Peter Stack of the San Francisco Chronicle wrote, "Wistful but not precious, ' The Land Girls ' takes a bit of acclimatizing, but Leland ('Wish You Were Here ') makes his leisurely pacing an attribute. The film sinks into the green, serene countryside, a place where emotions echo big."

Box office
After 11 weeks on release the film had grossed £1.3 million ($2.2 million) in the United Kingdom. It grossed $1 million internationally for a worldwide total of $3.2 million.

title: The Land Girls


text:
Talk:Voice acting

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 19 August 2019 and 13 December 2019. Further details are available on the course page. Student editor(s): Deltonkain.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 08:38, 18 January 2022 (UTC)

title: Talk:Voice acting


text:
Women's Land Army

The Women's Land Army (WLA) was a British civilian organisation created in 1917 by the Board of Agriculture during the First World War to bring women into work in agriculture, replacing men called up to the military. Women who worked for the WLA were commonly known as Land Girls (Land Lassies). The Land Army placed women with farms that needed workers, the farmers being their employers. The members picked crops and did all the labour to feed the country. Notable members include Joan Quennell, later a Member of Parliament; John Stewart Collis, Irish author and pioneer ecologist; the archaeologist Lily Chitty and the botanist Ethel Thomas. It was disbanded in 1919 but revived in June 1939 under the same name to again organise new workers to replace workers that served in the military during the Second World War.

First World War
The Women's Farm and Garden Union had existed since 1899 and in February 1916 they sent a deputation to meet Lord Selborne. Selborne's Ministry of Agriculture agreed to fund a Women’s National Land Service Corps with a grant of £150. Louise Wilkins was to lead the new organisation that was to focus on recruiting women for emergency war work. They were tasked with improving recruitment and provide propaganda about the good cause of women of all classes undertaking agricultural work. The new members of the organisation were to not become agricultural workers but to organise others (e.g. in villages) to do this work. By the end of 1916 they had recruited 2,000 volunteers but they estimated that 40,000 was required. At the Women’s National Land Service Corps's suggestion a Land Army was formed. The WNLSC continued to deal with recruitment and the network assisted in the launch of a "Land Army"; by April 1917 they had over 500 replies and 88 joined the new Land Army where they became group leaders and supervisors.

In time the Land Army would take on 23,000 workers who took the place of the 100,000 workers lost to the forces. The women were paid 18 shillings a week and this could be increased to 20 shillings (a pound) if they were considered efficient. 23,000 was a significant contribution but there were estimated to be 300,000 women working on the land during the First World War.

A Good Service Ribbon was awarded to eligible women. January 1918 saw the publication of the first issue of The Landswoman, the official monthly magazine of the Women’s Land Army and the Women’s Institutes. The organisation was disbanded in November 1919.

Second World War
As the prospect of war became increasingly likely, the government wanted to increase the amount of food grown within Britain. In April 1939, peacetime conscription was introduced for the first time in British history, which led to shortages of workers on the farms. To grow more food, more help was needed on the farms and so the government restarted the Women's Land Army in July 1939. Though under the Ministry of Agriculture and Fisheries, it was given an honorary head – Lady Denman. At first it asked for volunteers. This was supplemented by conscription, so that by 1944 it had over 80,000 members.

Inez Jenkins, who had served as Lady Deman's assistant director during the establishment of the WLA served as Chief Administrative Officer until 1948. The last Chief of the WLA was Amy Curtis. The WLA lasted until its official disbandment on 30 November 1950.

The majority of the Land Girls already lived in the countryside, but more than a third came from London and the industrial cities of the north of England. A separate branch was set up in 1942 for forestry industry work, officially known as the Women's Timber Corps and with its members colloquially known as "Lumber Jills" – this was disbanded in 1946.

In 1943, during the Second World War, Amelia King was refused work because she was black. The decision was overturned after being raised in the House of Commons by her MP, Walter Edwards.

Commemoration
In October 2012, the Prince of Wales unveiled the first memorial to the WLA of both World Wars, on the Fochabers estate in Moray, Scotland. The sculpture was designed by Peter Naylor. In October 2014, a memorial statue to the Women's Timber Corps and both incarnations of the Women's Land Army was unveiled at the National Memorial Arboretum in Staffordshire, England.

Recognition
In 1925 the Five Sisters window at York Minster was rededicated to the 1,513 women who died in the line of service during WWI, including ten women from the Land Army.

In December 2007, following campaigning by former Land Girl Hilda Gibson, the Department for Environment, Food and Rural Affairs (DEFRA) announced that the efforts of the Women's Land Army and the Women's Timber Corps would be formally recognised with the presentation of a specially designed commemorative badge to the surviving members. The badge of honour was awarded in July 2008 to over 45,000 former Land Girls.

In October 2012, the Prince of Wales unveiled the first memorial to the WLA of both World Wars, on the Fochabers estate in Moray, Scotland. The sculpture was designed by Peter Naylor. In October 2014, a memorial statue to the Women's Timber Corps and both incarnations of the Women's Land Army was unveiled at the National Memorial Arboretum in Staffordshire, England.

In popular culture
The Women's Land Army was the subject of:
 * The Land Girls (1998), a film loosely based on Angela Huth's book Land Girls (1995)
 * The ITV sitcom Backs to the Land (1977–78)
 * The BBC dramatic TV series Land Girls (2009–11)
 * The Powell and Pressburger 1944 film A Canterbury Tale features as the female lead a Land Girl, portrayed by Sheila Sim.

It also figured largely in:
 * Series 3, Episode 3 (2004) of the ITV detective series Foyle's War, entitled "They Fought in the Fields".
 * In the 2002 detective novel A Presumption of Death by Jill Paton Walsh and Dorothy L. Sayers, the plot centres on Harriet Vane and Lord Peter Wimsey trying to solve the murder of a land girl in Hertfordshire.
 * The Play For Today episode Rainy Day Women.

title: Women's Land Army


text:
Leisele

Leisele is a small Belgian town in Alveringem near the French border with about 750 inhabitants.

title: Leisele


text:
Peace and Freedom Party

The Peace and Freedom Party (PFP) is a left-wing political party with ballot status in California. Its first candidates appeared on the 1966 New York ballot. The Peace and Freedom Party of California was organized in early 1967, gathering over 103,000 registrants which qualified its ballot status in January 1968 under the California Secretary of State Report of Registration.

The party has appeared in other states as an antiwar and pro-civil rights organization opposed to the Vietnam War and supporting black liberation, farm-worker organizing, women's liberation, and the gay rights movement. Its presidential candidates were Leonard Peltier in 2004, Ralph Nader in 2008,  Roseanne Barr in 2012 and Gloria La Riva in 2016 and 2020.

Platform
According to its website, the party "is committed to feminism, socialism, democracy, ecology, and racial equality", advocating "to build a mass-based socialist party throughout the country". It is a strong advocate of environmentalism, aboriginal rights, rights to sexuality, government-funded health care, a woman's right to an abortion, public education, subsidized housing, and a socialist-run economy.

Founding
The Peace and Freedom Party grew out of the civil rights and anti-war movements on June 23, 1967. The major factors behind the new party were unhappiness with the Democratic Party's support for the war in Vietnam and also a perception that the Democrats were failing to effectively support the civil rights movement.

105,100 signatures were submitted on January 2, 1968, to receive party status in California. The party has had ballot access in California since 1968, except between 1998 and 2002.

In the 1966 House of Representatives elections, three people ran under the Peace and Freedom Party banner. Herbert Aptheker received 3,562 votes in New York state's 12th Congressional District; Robert B. Shaw received 1,974 votes in Washington state's 7th Congressional District, and Frank Patterson received 1,105 votes in Washington state's 2nd Congressional District. Late 1966 began a number of voter registration drives in various states with the intent to build a national party. Most notably in the San Francisco Bay Area and in Los Angeles county, activists became deputy registrars of voters as peace organizers, anti-war veterans and civil rights workers.

Election of 1968
The party achieved ballot status in California in January 1968 by registering over 105,000 voters under its banner. It later got ballot status in 13 other states, but in all of those, the election laws and small organization meant that most were unable to retain ballot status after 1968.

The PFP's first national convention to nominate candidates for president and vice president was held in Ann Arbor, Michigan on August 17–18, 1968. Ric Hyland, as California PFP's National Administrator, was convention chair. Eldridge Cleaver was nominated for president over Richard C. "Dick" Gregory by a margin of 161.5 to 54. Cleaver, a convicted felon and Black Panther spokesman, was technically not eligible to run since he would not yet be 35 by the time of the inauguration in January 1969. Due to the needs of the state parties to collect signatures, the party fielded several vice presidential nominees, including Chicago activist Peggy Terry, activist Rodolfo "Corky" Gonzales, radical economist Doug Dowd and Judith Mage, who had been nominated at the national convention. Cleaver personally preferred Yippie leader Jerry Rubin. Gregory also appeared on the ballot in several states along with his vice presidential running mate Mark Lane as the Peace and Freedom Party candidate as well as in New York as the candidate of the Freedom and Peace Party. Two states (California and Utah) refused to list Cleaver on the ballot, although each state listed the presidential electors and candidates for vice president (Peggy Terry in California and Corky Gonzales in Utah).

A variety of people joined the party in its first election. Bob Avakian was a spokesman for the party in the San Francisco Area and from the north coast where artists and activists such as Emmy Lou Packard and Byron Randall were involved. New York's Peace and Freedom Party consisted of a fractious coalition of competing Marxist groups, along with libertarians led by economist Murray Rothbard. Libertarians and some anarchists briefly competed for the leadership in the California party by running against the Socialist Campaign Collective and the Unity Coalition. The convention was deadlocked when the libertarians and about a third of the Unity Coalition walked out and formed a rump convention, leaving the socialists and their feminist allies with a clear majority. Most of the Libertarians left following the walkout at the 1974 convention. At that event, the California Secretary of State ruled that the group of people who had voted to make the party feminist and socialist would be considered the official party in California since they were the ones who had remained in the convention room.

In the election of 1968, the party fared fairly well for a newly ballot qualified party. Nationally, Gregory outpolled Cleaver, receiving 47,097 votes to Cleaver's 36,623 because some states had disqualified Cleaver because of his age. In California and Utah, where no presidential nominee appeared on the ballot, voters cast 27,887 votes for the party presidential race while the vice presidential candidate was on the ballot. The full nationwide vote for presidential electors was 111,607. Party candidates for the Senate received a national total of 105,411 votes. In Utah, the party fielded folk singer Bruce "U Utah" Phillips for Congress who trailed with 2,019 votes. The party retained ballot status in California in the 1970 general election, which it retained for a number of years except for 1999 to 2003. In 2003, Peace and Freedom Party became the first party in the history of California to regain its ballot status.

In 1968, the party held a statewide founding convention in Richmond, California. San Francisco chair Ric Hyland, co-founder of the Radical Caucus, nominated Eldridge Cleaver for president. Cleaver beat out Dick Gregory and Dr. Ben Spock on the basis of the slogan: We are not seeking the candidate with the broadest appeal, we seek the candidate with the deepest truth. Radical journalists Paul Jacobs and Robert Scheer were selected as the party's candidates for the U.S. Senate.

In 1970, Marge Buckley received 177,716 or 2.8% of the vote for Attorney General of California and C.T. Weber had 149,961 recorded votes (2.4%) in the vote for State Controller. By getting over 2 percent of the statewide vote, each of these candidates insured the party would maintain on the ballot through the 1974 election.

People's Party
After 1968, the party affiliates in most states dissolved primarily because of ballot access restrictions in many states. However, the California party continued to maintain enough registered voters to hold on to its ballot status and in some partisan districts the party held the balance of power between the so-called major parties.

Throughout the 1970s and into the 1980s, the California party continued to contest local elections and sometimes win city and service district elections, most notably in Sonoma County, where it won three of the five member Cotati city seats council. Another milestone was reached when Kayren Hudiburgh of San Francisco was elected state chair in 1974 of the California party, the first woman to hold this position in any of the ballot qualified parties in the state. Hudiburgh also ran twice for the state assembly in this time period.

The California party became part of the coalition making up the national left-wing People's Party. For 1972, the People's Party nominated the feminist and democratic socialist, the noted anti-war activist Benjamin Spock for president along with Julius Hobson of the D.C. Statehood Party for vice president. In 1976, the party nominated Margaret Wright as its first woman contender for president. Wright lived in the Watts section of Los Angeles and had worked closely with the Black Panther Party. Wright was also noted as being the founder of Women Against Racism.

The Internationalist Workers Party, New Alliance Party, and Socialist Party USA sought the party's ballot line during the 1988 presidential election. The factions within the PFP could not agree on a candidate and did not have a candidate for the election.

1998 ― Present
In 1998, the Peace and Freedom Party of California failed to attain more than the required two percent of the votes cast for one of its statewide candidates and was removed from the ballot as a ballot qualified party. In 2003, after a voter registration drive Peace and Freedom Party became the first ballot-qualified party in California history to lose its ballot status for more than one election and then requalify for the ballot. Longtime Peace and Freedom Party activist C. T. Weber was one of 135 candidates who ran for governor in the October 2003 recall election. In this recall, voters removed then-Governor Gray Davis (a Democrat) and elected Republican Arnold Schwarzenegger. At its August 2004 state convention, the Native American activist Leonard Peltier was nominated as Peace and Freedom Party's presidential candidate. Peltier was at the time (and still is) imprisoned serving a life term for the 1975 murders of two FBI agents; the Party considers him to be a political prisoner. Party members who supported Peltier's candidacy hoped to draw attention to his case and to the effort to win a presidential pardon for Peltier.

The party again fell under the required number of registered voters to retain ballot status in February 2006 and was declared disqualified by the California Secretary of State. However, citing previous instances in which parties not meeting the ballot qualification criteria were still allowed to participate in primary elections and the fact that there had not yet been a regular gubernatorial election since the party regained its ballot status (and as such, the decision was premature), the decision to bar the party from the June 2006 Primary was reversed after less than a week.

In the 2006 California elections, two statewide Peace and Freedom Party candidates received more than the required vote, thus ensuring the party's ballot status for another four years (Elizabeth Cervantes Barron received 212,383 votes, 2.5% of the total, for Controller; and Tom Condit received 187,618 votes, 2.2% of the total, for Insurance Commissioner)-

On the March 30, 2008, the State Central Committee endorsed a plan to create a National Organizing Committee (NOC) and a national political party. The NOC was instructed to work toward a national "multi-tendency non-sectarian organization committed to socialism, democracy, feminism, environmentalism and racial equality". A national organizing conference was set for December 2008 following the general election.

A political convention was held August 2–3, 2008 in Sacramento to select the party's 2008 presidential ticket. Contending for the nomination were Gloria La Riva (also nominee of the Party for Socialism and Liberation), Cynthia McKinney (also nominee of the Green Party) Brian Moore (also nominee of the Socialist Party), and independent presidential candidate Ralph Nader, who won. The results were the following: Nader (46%), Gloria La Riva (27%), Brian Moore (10%) and Cynthia McKinney (6%). Nader's running mate, former San Francisco Supervisor Matt Gonzalez, was endorsed for vice president by acclamation. The nomination ensured that the Nader/Gonzalez presidential ticket would appear on the ballot in California for the 2008 election.

On August 6, the Nader/Gonzalez campaign submitted sufficient signatures to appear on the Iowa and Utah ballots as the Peace and Freedom Party candidate. This was the first expansion of the party beyond California since the 1970s. However, the party did not achieve the votes necessary to guarantee ballot access in Iowa and Utah in subsequent elections.

Since 1968, over 400 different candidates have sought Peace and Freedom Party nominations for public office.

In 2016, the party's California state chair wrote the California Secretary of State, asking for Jill Stein to be placed on the party's ballot. Stein wrote as well. She was kept off the party's California primary ballot by the Secretary of State. The Peace and Freedom's 2016 presidential candidate Gloria La Riva was also the nominee of the Party for Socialism and Liberation.

California's 2018 gubernatorial primary had statewide office candidates registered in the party. In the race for Insurance Commissioner, Peace and Freedom candidate Nathalie Hrizi received 316,149 votes, 5.0% of the total.

In 2019, the Peace and Freedom Party in California grew from 76,784 registered voters in February to 90,121 in October. This growth of 17.4% was the highest growth rate achieved by any California political party. On more recent registration statistic reports, the party has had only increments of registrations, up to 105,535 registrations as of February 10, 2021.

In the 2022 California elections, the PFP joined the Green Party of California in creating the "Left Unity Slate," a set of candidates that were endorsed by both parties running for various offices across the state. The leader of the ticket, writer Luis J. Rodriguez, earned 1.8% of the vote in the gubernatorial primary on June 5. The most successful statewide candidate was Marco Amaral, who ran in the nonpartisan superintendent of public instruction election and earned 8.7% of the primary vote.

For the 2024 United States presidential election, the party plans to hold a non-binding preference primary in California on Super Tuesday, March 5. The party's presidential nominee will be chosen by the state central committee in August.

Participation of "independent" voters
Since January 2001, California has had a "modified" closed primary system in which political parties can determine whether or not to allow voters who are not affiliated with any party (i.e. "independent") to participate in the party's primary. The passage of Proposition 14 limited this "modified" closed primary system to primaries for President of the United States, starting with the 2012 primaries. Since the primaries in 2004, the Peace and Freedom Party has never allowed "independent" voters to vote in their party's primaries.

Presidential tickets
Notes:

Congressional elections

 * 2022, Jose Cortes	received 3,343 votes, or 2.2 percent, in the top-two primary for California's_51st_congressional_district.
 * 2022, John Parker received 105,477 votes, or 1.7 percent in the top-two primary for United States Senate.
 * 2018, John Parker ran on the PFP ballot in the top-two primary against incumbent Dianne Feinstein for United States Senate. Parker received 12,606	votes.
 * 1968 – 7th California's congressional district, Huey P. Newton: 12,164 votes (7.5%)

title: Peace and Freedom Party


text:
Group cohomology

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of $$G^n$$ representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups $$H^n(G,M)$$. The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients.

These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group G is the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of $$\Z$$ can be thought of as the singular cohomology of the circle S1. Likewise, the group cohomology of $$\Z/2\Z$$ is the singular cohomology of $$\mathbb{P}^{\infty}(\R).$$

A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.

Motivation
A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M. We will write G multiplicatively and M additively.

Given such a G-module M, it is natural to consider the submodule of G-invariant elements:

 * $$ M^{G} = \lbrace x \in M \ | \ \forall g \in G : \ gx=x \rbrace. $$

Now, if N is a G-submodule of M (i.e., a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in $$M/N$$ are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The purpose of the first group cohomology $$H^1(G,N)$$ is to precisely measure this difference.

The group cohomology functors $$H^*$$ in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence.

Definitions
The collection of all G-modules is a category (the morphisms are equivariant group homomorphisms, that is group homomorphisms f with the property $$f(gx) = g(f(x))$$ for all g in G and x in M). Sending each module M to the group of invariants $$M^G$$ yields a functor from the category of G-modules to the category Ab of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors. Their values are abelian groups and they are denoted by $$H^n(G,M)$$, "the n-th cohomology group of G with coefficients in M". Furthermore, the group $$H^0(G,M)$$ may be identified with $$M^G$$.

Cochain complexes
The definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful. For $$n \ge 0,$$ let $$C^n(G,M)$$ be the group of all functions from $$G^n$$ to M (here $$G^0$$ means $$\operatorname{id}_G$$). This is an abelian group; its elements are called the (inhomogeneous) n-cochains. The coboundary homomorphisms are defined by

 * $$\begin{cases}

d^{n+1} \colon C^n (G,M) \to C^{n+1}(G,M) \\ \left(d^{n+1}\varphi\right) (g_1, \ldots, g_{n+1}) = g_1\varphi(g_2, \dots, g_{n+1}) + \sum_{i=1}^n (-1)^i \varphi \left (g_1,\ldots, g_{i-1}, g_ig_{i+1}, \ldots, g_{n+1} \right ) + (-1)^{n+1}\varphi(g_1,\ldots, g_n). \end{cases}$$

One may check that $$d^{n+1} \circ d^n = 0,$$ so this defines a cochain complex whose cohomology can be computed. It can be shown that the above-mentioned definition of group cohomology in terms of derived functors is isomorphic to the cohomology of this complex

 * $$H^n(G,M) = Z^n(G,M)/B^n(G,M).$$

Here the groups of n-cocycles, and n-coboundaries, respectively, are defined as

 * $$Z^n(G,M) = \ker(d^{n+1}) $$
 * $$B^n(G,M) = \begin{cases} 0 & n = 0 \\ \operatorname{im}(d^{n}) & n \geqslant 1 \end{cases}$$

The functors Extn and formal definition of group cohomology
Interpreting G-modules as modules over the group ring $$\Z[G],$$ one can note that

 * $$H^{0}(G,M) = M^G = \operatorname{Hom}_{\Z[G]}(\Z ,M),$$

i.e., the subgroup of G-invariant elements in M is identified with the group of homomorphisms from $$\Z$$, which is treated as the trivial G-module (every element of G acts as the identity) to M.

Therefore, as Ext functors are the derived functors of Hom, there is a natural isomorphism

 * $$H^{n}(G,M) = \operatorname{Ext}^{n}_{\Z [G]}(\Z, M).$$

These Ext groups can also be computed via a projective resolution of $$\Z$$, the advantage being that such a resolution only depends on G and not on M. We recall the definition of Ext more explicitly for this context. Let F be a projective $\Z[G]$-resolution (e.g. a free $\Z[G]$-resolution) of the trivial $$\Z[G]$$-module $$\Z$$:

 * $$ \cdots \to F_n\to F_{n-1} \to\cdots \to F_0\to \Z\to 0.$$

e.g., one may always take the resolution of group rings, $$F_n = \Z[G^{n+1}],$$ with morphisms

 * $$\begin{cases}f_n : \Z[G^{n+1}] \to \Z[G^n] \\ (g_0, g_1, \ldots, g_n) \mapsto \sum_{i=0}^n (-1)^i \left (g_0, \ldots, \widehat{g_i}, \dots, g_n \right ) \end{cases}$$

Recall that for $$\Z[G]$$-modules N and M, HomG(N, M) is an abelian group consisting of $$\Z[G]$$-homomorphisms from N to M. Since $$\operatorname{Hom}_{G}(-,M)$$ is a contravariant functor and reverses the arrows, applying $$\operatorname{Hom}_{G}(-,M)$$ to F termwise and dropping $$\operatorname{Hom}_G(\Z, M)$$ produces a cochain complex $$\operatorname{Hom}_{G}(-,M)(F,M)$$:

 * $$\cdots \leftarrow \operatorname{Hom}_G(F_n,M)\leftarrow \operatorname{Hom}_G(F_{n-1},M) \leftarrow \dots \leftarrow \operatorname{Hom}_G (F_0,M) \leftarrow 0.$$

The cohomology groups $$H^*(G,M)$$ of G with coefficients in the module M are defined as the cohomology of the above cochain complex:

 * $$ H^n(G,M)=H^n({\rm Hom}_{G}(F,M)), \qquad n \geqslant 0.$$

This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the elements of $$\operatorname{Hom}_G(F,M)$$, that is, functions $$\phi_n\colon G^n \to M$$ that obey

 * $$ g\phi_n(g_1,g_2,\ldots, g_n)= \phi_n(gg_1,gg_2,\ldots, gg_n).$$

The coboundary operator $$\delta\colon C^n \to C^{n+1}$$ is now naturally defined by, for example,

 * $$ \delta \phi_2(g_1, g_2,g_3)= \phi_2(g_2,g_3)-\phi_2(g_1,g_3)+ \phi_2(g_1,g_2).$$

The relation to the coboundary operator d that was defined in the previous section, and which acts on the "inhomogeneous" cochains $$ \varphi$$, is given by reparameterizing so that

 * $$\begin{align}

\varphi_2(g_1,g_2) &= \phi_3(1, g_1,g_1g_2) \\ \varphi_3(g_1,g_2,g_3) &= \phi_4(1, g_1,g_1g_2, g_1g_2g_3), \end{align}$$

and so on. Thus

 * $$\begin{align}

d \varphi_2(g_1,g_2,g_3) &= \delta \phi_3(1,g_1, g_1g_2,g_1g_2g_3)\\ & = \phi_3(g_1, g_1g_2,g_1g_2g_3) - \phi_3(1, g_1g_2, g_1g_2g_3) +\phi_3(1,g_1, g_1g_2g_3) - \phi_3(1,g_1,g_1g_2) \\ & = g_1\phi_3(1, g_2,g_2g_3) - \phi_3(1, g_1g_2, g_1g_2g_3) +\phi_3(1,g_1, g_1g_2g_3) - \phi_3(1,g_1,g_1g_2) \\ & = g_1\varphi_2(g_2,g_3) -\varphi_2(g_1g_2,g_3)+\varphi_2(g_1,g_2g_3) -\varphi_2(g_1,g_2), \end{align}$$

as in the preceding section.

Group homology
Dually to the construction of group cohomology there is the following definition of group homology: given a G-module M, set DM to be the submodule generated by elements of the form g·m − m, g ∈ G, m ∈ M. Assigning to M its so-called coinvariants, the quotient

 * $$M_G:=M/DM,$$

is a right exact functor. Its left derived functors are by definition the group homology

 * $$H_n(G,M).$$

The covariant functor which assigns MG to M is isomorphic to the functor which sends M to $$\Z \otimes_{\Z[G]} M,$$ where $$\Z$$ is endowed with the trivial G-action. Hence one also gets an expression for group homology in terms of the Tor functors,

 * $$H_n(G,M) = \operatorname{Tor}_n^{\Z[G]}(\Z,M)$$

Note that the superscript/subscript convention for cohomology/homology agrees with the convention for group invariants/coinvariants, while which is denoted "co-" switches:
 * superscripts correspond to cohomology H* and invariants XG while
 * subscripts correspond to homology H∗ and coinvariants XG := X/G.

Specifically, the homology groups Hn(G, M) can be computed as follows. Start with a projective resolution F of the trivial $$\Z[G]$$-module $$\Z,$$ as in the previous section. Apply the covariant functor $$\cdot \otimes_{\Z[G]} M$$ to F termwise to get a chain complex $$F \otimes_{\Z[G]} M$$:

 * $$ \cdots \to F_n\otimes_{\Z[G]}M\to F_{n-1}\otimes_{\Z[G]}M \to\cdots \to F_0\otimes_{\Z[G]}M\to \Z\otimes_{\Z[G]}M.$$

Then Hn(G, M) are the homology groups of this chain complex, $$H_n(G,M)=H_n(F\otimes_{\Z[G]}M)$$ for n ≥ 0.

Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups.

The group homology $$H_*(G, k)$$ of abelian groups G with values in a principal ideal domain k is closely related to the exterior algebra $$\wedge^* (G \otimes k)$$.

H 1
The first cohomology group is the quotient of the so-called crossed homomorphisms, i.e. maps (of sets) f : G → M satisfying f(ab) = f(a) + af(b) for all a, b in G, modulo the so-called principal crossed homomorphisms, i.e. maps f : G → M given by f(g) = gm−m for some fixed m ∈ M. This follows from the definition of cochains above.

If the action of G on M is trivial, then the above boils down to H1(G,M) = Hom(G, M), the group of group homomorphisms G → M, since the crossed homomorphisms are then just ordinary homomorphisms and the coboundaries (i.e. the principal crossed homomorphisms) must have image identically zero: hence there is only the zero coboundary.

On the other hand, consider the case of $$H^1(\Z/2, \Z_-),$$ where $$\Z_-$$ denotes the non-trivial $$\Z/2$$-structure on the additive group of integers, which sends a to -a for every $$a \in \Z $$; and where we regard $$\Z/2$$ as the group $$\{ \pm 1 \}$$. By considering all possible cases for the images of $$\{ 1,-1 \}$$, it may be seen that crossed homomorphisms constitute all maps $$f_t: \{ \pm 1 \} \to \Z$$ satisfying $$f_t(1) = 0$$ and $$f_t(-1) = t$$ for some arbitrary choice of integer t. Principal crossed homomorphisms must additionally satisfy $$f_t(-1) = (-1)*m - m = -2m$$ for some integer m: hence every crossed homomorphism $$f_t$$ sending -1 to an even integer $$t = -2m$$ is principal, and therefore:

 * $$H^1(\Z/2,\Z_{-})\cong \Z/2 = {\rm\ (say)\ \it} \langle f: f(1)=0, f(-1)=1\rangle,$$

with the group operation being pointwise addition: $$(f_s+f_t)(x) = f_s(x) + f_t(x) = f_{s+t}(x)$$, noting that $$f_0$$ is the identity element.

H 2
If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H2(G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H2(G,M) classifies the isomorphism classes of all extensions $$0 \to M \to E \to G \to 0$$ of G by M, in which the action of G on E (by inner automorphisms), endows (the image of) M with an isomorphic G-module structure.

In the example from the section on $$H^1$$ immediately above, $$H^2(\Z/2, \Z_-) =0,$$ as the only extension of $$\Z/2$$ by $$\Z$$ with the given nontrivial action is the infinite dihedral group, which is a split extension and so trivial inside the $$H^2$$ group. This is in fact the significance in group-theoretical terms of the unique non-trivial element of $$H^1(\Z/2, \Z_-),$$.

An example of a second cohomology group is the Brauer group: it is the cohomology of the absolute Galois group of a field k which acts on the invertible elements in a separable closure:

 * $$H^2\left(\mathrm{Gal}(k), (k^\mathrm{sep})^\times\right).$$

See also.

Group cohomology of a finite cyclic group
For the finite cyclic group $$G=C_m$$ of order $$m$$ with generator $$\sigma$$, the element $$\sigma -1 \in \mathbb{Z}[G]$$ in the associated group ring is a divisor of zero because its product with $$N$$, given by"$N = 1 + \sigma + \sigma^2 + \cdots + \sigma^{m-1} \in \mathbb{Z}[G],$"gives $$\begin{align} N(1-\sigma) &= 1 + \sigma + \cdots + \sigma^{m-1} \\ &\quad- \sigma - \sigma^2 - \cdots - \sigma^{m} \\ &=1 - \sigma^m \\ &= 0. \end{align}$$ This property can be used to construct the resolution of the trivial $$\mathbb{Z}[G]$$-module $$\mathbb{Z}$$ via the complex"$\cdots \xrightarrow{\sigma - 1}\mathbb{Z}[G] \xrightarrow{N} \mathbb{Z}[G] \xrightarrow{\sigma - 1}\mathbb{Z}[G] \xrightarrow{\text{aug}} \mathbb{Z} \to 0$|undefined"giving the group cohomology computation for any $$\mathbb{Z}[G]$$-module $$A$$. Note the augmentation map gives the trivial module $$\mathbb{Z}$$ its $$\mathbb{Z}[G]$$-structure by"$\text{aug}\left(\sum_{g \in G}a_gg \right) = \sum_{g \in G}a_g$"This resolution gives a computation of the group cohomology since there is the isomorphism of cohomology groups"$H^k(G,A) \cong \text{Ext}^k_{\mathbb{Z}[G]}(\mathbb{Z},A)$"showing that applying the functor $$\text{Hom}_{\mathbb{Z}[G]}(-,A)$$ to the complex above (with $$\mathbb{Z}$$ removed since this resolution is a quasi-isomorphism), gives the computation $$H^k(G,A) = \begin{cases} A^G/NA & k\text{ even}, k \geq 2 \\ {}_NA/(\sigma - 1)A & k\text{ odd}, k \geq 1 \end{cases}$$ for"${}_NA = \{a \in A : Na = 0\}$"For example, if $$A = \mathbb{Z}$$, the trivial module, then $$\mathbb{Z}^G = \mathbb{Z} $$, $$N\mathbb{Z} = \text{aug}(N)\mathbb{Z} = m\mathbb{Z}$$, and $${}_N\mathbb{Z} = 0$$, hence $$H^k(C_m,\mathbb{Z}) = \begin{cases} \mathbb{Z}/m\mathbb{Z} & k\text{ even}, k \geq 2 \\ 0 & k\text{ odd}, k \geq 1 \end{cases}$$

Explicit cocycles
Cocycles for the group cohomology of a cyclic group can be given explicitly using the Bar resolution. We get a complete set of generators of $$l$$-cocycles for $$l$$ odd as the maps"$\omega_a: B_l \to k^*$"given by $$[g^{i_1},\ldots, g^{i_l}] \mapsto \zeta_m^{ ai_1 \left[ \frac{i_2 + i_3}{m} \right] \cdots \left[ \frac{i_{l-1} + i_l}{m} \right] }$$ for $$l$$ odd, $$0 \leq a \leq m-1$$, $$\zeta_m$$ a primitive $$m$$-th root of unity, $$k$$ a field containing $$m$$-th roots of unity, and"$\left[\frac{a}{b} \right]$"for a rational number $$a/b$$ denoting the largest integer not greater than $$a/b$$. Also, we are using the notation"$B_l = \bigoplus_{0 \leq i_1,\ldots, i_l \leq m-1}\mathbb{Z}G \cdot [g^{i_1},\ldots, g^{i_l}]$"where $$g$$ is a generator for $$G = C_m$$. Note that for $$l$$ non-zero even indices the cohomology groups are trivial.

Using a resolution
Given a set $$S$$ the associated free group $$G = \text{Free}(S) = \underset{s \in S}{*} \mathbb{Z}$$ has an explicit resolution of the trivial module $$\mathbb{Z}_{\text{triv}}$$ which can be easily computed. Notice the augmentation map"$\text{aug}:\mathbb{Z}[G] \to \mathbb{Z}_{\text{triv}}$|undefined"has kernel given by the free submodule $$I_S$$ generated by the set $$\{s - 1 : s \in S \}$$, so"$I_S = \bigoplus_{s \in S} \mathbb{Z}[G]\cdot (s-1)$."Because this object is free, this gives a resolution"$0 \to I_S \to \mathbb{Z}[G] \to \mathbb{Z}_{\text{triv}} \to 0$|undefined"hence the group cohomology of $$G$$ with coefficients in $$\mathbb{Z}_{\text{triv}}$$ can be computed by applying the functor $$\text{Hom}_{\mathbb{Z}[G]}(-,\mathbb{Z})$$ to the complex $$0 \to I_S \to \mathbb{Z}[G] \to 0$$, giving $$H^k(G,\mathbb{Z}_{\text{triv}}) = \begin{cases} \mathbb{Z} & k = 0 \\ \bigoplus_{s \in S}\mathbb{Z} & k = 1 \\ 0 & k \geq 2 \end{cases}$$ this is because the dual map $$\text{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}[G],\mathbb{Z}_{\text{triv}}) \to \text{Hom}_{\mathbb{Z}[G]}(I_S,\mathbb{Z}_{\text{triv}})$$ sends any $$\mathbb{Z}[G]$$-module morphism"$\phi:\mathbb{Z}[G] \to \mathbb{Z}_{\text{triv}}$|undefined"to the induced morphism on $$I_S$$ by composing the inclusion. The only maps which are sent to $$0$$ are $$\mathbb{Z}$$-multiples of the augmentation map, giving the first cohomology group. The second can be found by noticing the only other maps"$\psi \in \text{Hom}_{\mathbb{Z}[G]}(I_S,\mathbb{Z}_{\text{triv}})$|undefined"can be generated by the $$\mathbb{Z}$$-basis of maps sending $$(s-1) \mapsto 1$$ for a fixed $$s \in S$$, and sending $$(s'-1) \mapsto 0$$ for any $$s' \in S - \{s\}$$.

Using topology
The group cohomology of free groups $$\mathbb{Z}*\mathbb{Z}*\cdots *\mathbb{Z}$$ generated by $$n$$ letters can be readily computed by comparing group cohomology with its interpretation in topology. Recall that for every group $$G$$ there is a topological space $$BG$$, called the classifying space of the group, which has the property"$\pi_1(BG) = G \text{ and } \pi_k(BG) = 0 \text{ for } k \geq 2$"In addition, it has the property that its topological cohomology is isomorphic to group cohomology"$H^k(BG,\mathbb{Z}) \cong H^k(G,\mathbb{Z})$"giving a way to compute some group cohomology groups. Note $$\mathbb{Z}$$ could be replaced by any local system $$\mathcal{L}$$ which is determined by a map"$\pi_1(G) \to GL(V)$"for some abelian group $$V$$. In the case of $$B(\mathbb{Z}*\cdots *\mathbb{Z})$$ for $$n$$ letters, this is represented by a wedge sum of $$n$$ circles $$S^1 \vee \cdots \vee S^1$$ which can be showed using the Van-Kampen theorem, giving the group cohomology $$H^k(\mathbb{Z}*\cdots * \mathbb{Z}) = \begin{cases} \mathbb{Z} & k = 0 \\ \mathbb{Z}^n & k = 1 \\ 0 & k \geq 2 \end{cases}$$

Group cohomology of an integral lattice
For an integral lattice $$\Lambda$$ of rank $$n$$ (hence isomorphic to $$\mathbb{Z}^n$$), its group cohomology can be computed with relative ease. First, because $$B\mathbb{Z} \cong S^1$$, and $$B\mathbb{Z}\times B\mathbb{Z}$$ has $$\pi_1 \cong \mathbb{Z}\times\mathbb{Z}$$, which as abelian groups are isomorphic to $$\mathbb{Z}\oplus\mathbb{Z}$$, the group cohomology has the isomorphism"$H^k(\Lambda,\mathbb{Z}_{\text{triv}}) \cong H^k(\mathbb{R}^n/\mathbb{Z}^n,\mathbb{Z})$|undefined"with the integral cohomology of a torus of rank $$n$$.

Properties
In the following, let M be a G-module.

Long exact sequence of cohomology
In practice, one often computes the cohomology groups using the following fact: if

 * $$ 0 \to L \to M \to N \to 0 $$

is a short exact sequence of G-modules, then a long exact sequence is induced:

 * $$0\longrightarrow L^G \longrightarrow M^G \longrightarrow N^G \overset{\delta^0}{\longrightarrow} H^1(G,L) \longrightarrow H^1(G,M) \longrightarrow H^1(G,N) \overset{\delta^1}{\longrightarrow} H^2(G,L)\longrightarrow \cdots$$

The so-called connecting homomorphisms,

 * $$\delta^n : H^n (G,N) \to H^{n+1}(G, L)$$

can be described in terms of inhomogeneous cochains as follows. If $$c \in H^n(G, N)$$ is represented by an n-cocycle $$\phi: G^n \to N,$$ then $$\delta^n(c)$$ is represented by $$d^n(\psi),$$ where $$\psi$$ is an n-cochain $$G^n \to M$$ "lifting" $$\phi$$ (i.e. $$\phi$$ is the composition of $$\psi$$ with the surjective map M → N).

Functoriality
Group cohomology depends contravariantly on the group G, in the following sense: if f : H → G is a group homomorphism, then we have a naturally induced morphism Hn(G, M) → Hn(H, M) (where in the latter, M is treated as an H-module via f). This map is called the restriction map. If the index of H in G is finite, there is also a map in the opposite direction, called transfer map,

 * $$cor_H^G : H^n(H, M) \to H^n (G, M).$$

In degree 0, it is given by the map

 * $$\begin{cases} M^H \to M^G \\ m \mapsto \sum_{g \in G/H} gm \end{cases}$$

Given a morphism of G-modules M → N, one gets a morphism of cohomology groups in the Hn(G, M) → Hn(G, N).

Products
Similarly to other cohomology theories in topology and geometry, such as singular cohomology or de Rham cohomology, group cohomology enjoys a product structure: there is a natural map called cup product:

 * $$H^n(G, N) \otimes H^m(G, M) \to H^{n+m} (G, M \otimes N)$$

for any two G-modules M and N. This yields a graded anti-commutative ring structure on $$\oplus_{n \geqslant 0} H^n(G, R),$$ where R is a ring such as $$\Z$$ or $$\Z/p.$$ For a finite group G, the even part of this cohomology ring in characteristic p, $$\oplus_{n \geqslant 0} H^{2n}(G, \Z/ p)$$ carries a lot of information about the group the structure of G, for example the Krull dimension of this ring equals the maximal rank of an abelian subgroup $$(\Z / p)^r$$.

For example, let G be the group with two elements, under the discrete topology. The real projective space $$\mathbb{P}^{\infty}(\R)$$ is a classifying space for G. Let $$k=\mathbb{F}_2,$$ the field of two elements. Then

 * $$H^*(G;k)\cong k[x],$$

a polynomial k-algebra on a single generator, since this is the cellular cohomology ring of $$\mathbb{P}^{\infty}(\R).$$

Künneth formula
If, M = k is a field, then H*(G; k) is a graded k-algebra and the cohomology of a product of groups is related to the ones of the individual groups by a Künneth formula:

 * $$H^*(G_1\times G_2;k)\cong H^*(G_1;k)\otimes H^*(G_2;k).$$

For example, if G is an elementary abelian 2-group of rank r, and $$k=\mathbb{F}_2,$$ then the Künneth formula shows that the cohomology of G is a polynomial k-algebra generated by r classes in H1(G; k).,

 * $$H^*(G;k)\cong k[x_1, \ldots, x_r].$$

Homology vs. cohomology
As for other cohomology theories, such as singular cohomology, group cohomology and homology are related to one another by means of a short exact sequence

 * $$0 \to \mathrm{Ext}^1_{\Z}\left(H_{n-1}(G, \Z), A\right) \to H^n(G, A) \to \mathrm{Hom}\left(H_n(G, \Z), A\right) \to 0,$$

where A is endowed with the trivial G-action and the term at the left is the first Ext group.

Amalgamated products
Given a group A which is the subgroup of two groups G1 and G2, the homology of the amalgamated product $$G := G_1 \star_A G_2$$ (with integer coefficients) lies in a long exact sequence

 * $$\cdots \to H_n (A) \to H_n (G_1) \oplus H_n (G_2) \to H_n (G) \to H_{n-1}(A) \to \cdots$$

The homology of $$\mathrm{SL}_2(\Z) = \Z / 4 \star_{\Z/2} \Z/6$$ can be computed using this:

 * $$H_n(\mathrm{SL}_2(\Z)) = \begin{cases} \Z & n =0 \\ \Z/12 & \text{odd degrees} \\ 0 & \text{otherwise} \end{cases}$$

This exact sequence can also be applied to show that the homology of the $$\mathrm{SL}_2(k[t])$$ and the special linear group $$\mathrm{SL}_2(k)$$ agree for an infinite field k.

Change of group
The Hochschild–Serre spectral sequence relates the cohomology of a normal subgroup N of G and the quotient G/N to the cohomology of the group G (for (pro-)finite groups G). From it, one gets the inflation-restriction exact sequence.

Cohomology of the classifying space
Group cohomology is closely related to topological cohomology theories such as sheaf cohomology, by means of an isomorphism

 * $$H^n (BG, \Z) \cong H^n (G, \Z).$$

The expression $$BG$$ at the left is a classifying space for $$G$$. It is an Eilenberg–MacLane space $$K(G,1)$$, i.e., a space whose fundamental group is $$G$$ and whose higher homotopy groups vanish). Classifying spaces for $$\Z, \Z/2$$ and $$\Z/n$$ are the 1-sphere S1, infinite real projective space $$\mathbb{P}^{\infty}(\R) = \cup_n \mathbb{P}^n(\R),$$ and lens spaces, respectively. In general, $$BG$$ can be constructed as the quotient $$EG/G$$, where $$EG$$ is a contractible space on which $$G$$ acts freely. However, $$BG$$ does not usually have an easily amenable geometric description.

More generally, one can attach to any $$G$$-module $$M$$ a local coefficient system on $$BG$$ and the above isomorphism generalizes to an isomorphism

 * $$H^n (BG, M) = H^n (G, M).$$

Semi-direct products of groups
There is a way to compute the semi-direct product of groups using the topology of fibrations and properties of Eilenberg-Maclane spaces. Recall that for a semi-direct product of groups $$G = N \rtimes H$$ there is an associated short exact sequence of groups"$1 \to N \to N\rtimes H \to H \to 1$"Using the associated Eilenberg-Maclane spaces there is a Serre fibration"$K(N,1) \to K(G,1) \to K(H,1)$"which can be put through a Serre spectral sequence. This gives an $$E_2$$-page"$E_2^{p,q} = H^p(K(H,1),H^q(K(N,1))) \Rightarrow H^{p+q}(K(G,1))$"which gives information about the group cohomology of $$G$$ from the group cohomology groups of $$H,N$$. Note this formalism can be applied in a purely group-theoretic manner using the Lyndon–Hochschild–Serre spectral sequence.

Higher cohomology groups are torsion
The cohomology groups Hn(G, M) of finite groups G are all torsion for all n≥1. Indeed, by Maschke's theorem the category of representations of a finite group is semi-simple over any field of characteristic zero (or more generally, any field whose characteristic does not divide the order of the group), hence, viewing group cohomology as a derived functor in this abelian category, one obtains that it is zero. The other argument is that over a field of characteristic zero, the group algebra of a finite group is a direct sum of matrix algebras (possibly over division algebras which are extensions of the original field), while a matrix algebra is Morita equivalent to its base field and hence has trivial cohomology.

If the order of G is invertible in a G-module M (for example, if M is a $$\Q$$-vector space), the transfer map can be used to show that $$H^n(G,M) =0$$ for $$n \geqslant 1.$$ A typical application of this fact is as follows: the long exact cohomology sequence of the short exact sequence (where all three groups have a trivial G-action)

 * $$0 \to \Z \to \Q \to \Q / \Z \to 0$$

yields an isomorphism

 * $$\mathrm{Hom}(G, \Q / \Z) = H^1(G, \Q /\Z) \cong H^2(G, \Z).$$

Tate cohomology
Tate cohomology groups combine both homology and cohomology of a finite group G:

 * $$\widehat H^n(G, M) := \begin{cases} H^n(G, M) & n \geqslant 1 \\ \operatorname{coker} N & n=0 \\ \ker N & n = -1 \\ H_{-n-1}(G, M) & n \leqslant -2, \end{cases} $$

where $$N: M_G \to M^G$$ is induced by the norm map:

 * $$\begin{cases} M \to M \\ m \mapsto \sum_{g \in G} gm \end{cases}$$

Tate cohomology enjoys similar features, such as long exact sequences, product structures. An important application is in class field theory, see class formation.

Tate cohomology of finite cyclic groups, $$G = \Z/n,$$ is 2-periodic in the sense that there are isomorphisms

 * $$\widehat H^m(G, M) \cong \widehat H^{m+2}(G, M) \qquad \text{for all } m \in \Z.$$

A necessary and sufficient criterion for a d-periodic cohomology is that the only abelian subgroups of G are cyclic. For example, any semi-direct product $$\Z / n \rtimes \Z /m $$ has this property for coprime integers n and m.

Algebraic K-theory and homology of linear groups
Algebraic K-theory is closely related to group cohomology: in Quillen's +-construction of K-theory, K-theory of a ring R is defined as the homotopy groups of a space $$\mathrm{BGL}(R)^+.$$ Here $$\mathrm{GL}(R) = \cup_{n \ge 1} \mathrm{GL}_n(R)$$ is the infinite general linear group. The space $$\mathrm{BGL}(R)^+$$ has the same homology as $$\mathrm{BGL}(R),$$ i.e., the group homology of GL(R). In some cases, stability results assert that the sequence of cohomology groups

 * $$\dots \to H_m\left(\mathrm{GL}_n (R)\right) \to H_m\left(\mathrm{GL}_{n+1}(R)\right) \to \cdots$$

becomes stationary for large enough n, hence reducing the computation of the cohomology of the infinite general linear group to the one of some $$\mathrm{GL}_n(R)$$. Such results have been established when R is a field or for rings of integers in a number field.

The phenomenon that group homology of a series of groups $$G_n$$ stabilizes is referred to as homological stability. In addition to the case $$G_n = \mathrm{GL}_n(R)$$ just mentioned, this applies to various other groups such as symmetric groups or mapping class groups.

Projective representations and group extensions
In quantum mechanics we often have systems with a symmetry group $$G.$$ We expect an action of $$G$$ on the Hilbert space $$\mathcal{H}$$ by unitary matrices $$U(g).$$ We might expect $$U(g_1) U(g_2)= U(g_1g_2),$$ but the rules of quantum mechanics only require

 * $$U(g_1) U(g_2)= \exp \{2\pi i\omega(g_1,g_2)\} U(g_1g_2),$$

where $$\exp\{2\pi i\omega(g_1,g_2)\}\in{\rm U}(1)$$ is a phase. This projective representation of $$G$$ can also be thought of as a conventional representation of a group extension $$\tilde G$$ of $$G$$ by $$\mathrm{U}(1),$$ as described by the exact sequence

 * $$1 \to {\rm U}(1) \to \tilde G \to G\to 1.$$

Requiring associativity

 * $$U(g_1)[U(g_2)U(g_3)]= [U(g_1)U(g_2)]U(g_3)$$

leads to

 * $$\omega(g_2, g_3)-\omega(g_1g_2, g_3)+ \omega(g_1,g_2g_3)-\omega(g_1,g_2)=0,$$

which we recognise as the statement that $$d\omega(g_1,g_2,g_3)=0,$$ i.e. that $$\omega$$ is a cocycle taking values in $$\R/\Z\simeq {\rm U}(1).$$ We can ask whether we can eliminate the phases by redefining

 * $$U(g)\to \exp\{2\pi i\eta(g)\} U(g)$$

which changes

 * $$\omega(g_1,g_2) \to \omega(g_1,g_2) + \eta (g_2)- \eta(g_1g_2)+\eta(g_1).$$

This we recognise as shifting $$\omega$$ by a coboundary $$\omega \to \omega+d\eta.$$ The distinct projective representations are therefore classified by $$H^2(G, \R/\Z).$$ Note that if we allow the phases themselves to be acted on by the group (for example, time reversal would complex-conjugate the phase), then the first term in each of the coboundary operations will have a $$g_1$$ acting on it as in the general definitions of coboundary in the previous sections. For example, $$d\eta(g_1,g_2) \to g_1\eta(g_2)-\eta(g_1g_2)+\eta(g_1).$$

Cohomology of topological groups
Given a topological group G, i.e., a group equipped with a topology such that product and inverse are continuous maps, it is natural to consider continuous G-modules, i.e., requiring that the action

 * $$G \times M \to M$$

is a continuous map. For such modules, one can again consider the derived functor of $$M \mapsto M^G$$. A special case occurring in algebra and number theory is when G is profinite, for example the absolute Galois group of a field. The resulting cohomology is called Galois cohomology.

Non-abelian group cohomology
Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G.

The zeroth cohomology of G with coefficients in A is defined to be the subgroup

 * $$H^{0}(G,A)=A^{G},$$

of elements of A fixed by G.

The first cohomology of G with coefficients in A is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map $$\varphi$$ to be a 1-cocycle is that $$\varphi(gh) = \varphi(g)[g\varphi(h)]$$ and $$\ \varphi\sim \varphi'$$ if there is an a in A such that $$\ a\varphi'(g)=\varphi(g)\cdot(ga)$$. In general, $$H^1(G,A)$$ is not a group when A is non-abelian. It instead has the structure of a pointed set – exactly the same situation arises in the 0th homotopy group, $$\ \pi_0(X;x)$$ which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point.

Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let

 * $$1\to A\to B\to C\to 1\,$$

be a short exact sequence of G-groups, then there is an exact sequence of pointed sets

 * $$1\to A^G\to B^G\to C^G\to H^1(G,A) \to H^1(G,B) \to H^1(G,C).\,$$

History and relation to other fields
The low-dimensional cohomology of a group was classically studied in other guises, well before the notion of group cohomology was formulated in 1943–45. The first theorem of the subject can be identified as Hilbert's Theorem 90 in 1897; this was recast into Emmy Noether's equations in Galois theory (an appearance of cocycles for $$H^1$$). The idea of factor sets for the extension problem for groups (connected with $$H^2$$) arose in the work of Otto Hölder (1893), in Issai Schur's 1904 study of projective representations, in Otto Schreier's 1926 treatment, and in Richard Brauer's 1928 study of simple algebras and the Brauer group. A fuller discussion of this history may be found in.

In 1941, while studying $$H^2(G,\Z)$$ (which plays a special role in groups), Heinz Hopf discovered what is now called Hopf's integral homology formula, which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group:

 * $$ H_2(G,\Z) \cong (R \cap [F, F])/[F, R],$$

where $$G\cong F/R$$ and F is a free group.

Hopf's result led to the independent discovery of group cohomology by several groups in 1943-45: Samuel Eilenberg and Saunders Mac Lane in the United States ; Hopf and Beno Eckmann in Switzerland; Hans Freudenthal in the Netherlands ; and Dmitry Faddeev in the Soviet Union. The situation was chaotic because communication between these countries was difficult during World War II.

From a topological point of view, the homology and cohomology of G was first defined as the homology and cohomology of a model for the topological classifying space BG as discussed above. In practice, this meant using topology to produce the chain complexes used in formal algebraic definitions. From a module-theoretic point of view this was integrated into the Cartan–Eilenberg theory of homological algebra in the early 1950s.

The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. In turn, this led to the notion of Galois cohomology and étale cohomology (which builds on it). Some refinements in the theory post-1960 have been made, such as continuous cocycles and John Tate's redefinition, but the basic outlines remain the same. This is a large field, and now basic in the theories of algebraic groups.

The analogous theory for Lie algebras, called Lie algebra cohomology, was first developed in the late 1940s, by Claude Chevalley and Eilenberg, and Jean-Louis Koszul. It is formally similar, using the corresponding definition of invariant for the action of a Lie algebra. It is much applied in representation theory, and is closely connected with the BRST quantization of theoretical physics.

Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of spontaneous symmetry breaking phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter—short-range entangled states with symmetry. Short-range entangled states with symmetry are also known as symmetry-protected topological states.

title: Group cohomology