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Carson City, Nevada

Carson City is an independent city and the capital of the U.S. state of Nevada. As of the 2020 census, the population was 58,639, making it the 6th most populous city in the state. The majority of the city's population lives in Eagle Valley, on the eastern edge of the Carson Range, a branch of the Sierra Nevada, about 30 mi south of Reno. The city is named after the mountain man Kit Carson. The town began as a stopover for California-bound immigrants, but developed into a city with the Comstock Lode, a silver strike in the mountains to the northeast. The city has served as Nevada's capital since statehood in 1864; for much of its history it was a hub for the Virginia and Truckee Railroad, although the tracks were removed in 1950.

Before 1969, Carson City was the county seat of Ormsby County. That year, after a referendum approved merging the city and the county, the state legislature issued a revised city charter that merged them into the Consolidated Municipality of Carson City. With the consolidation, the city limits extend west across the Sierra Nevada to the California state line in the middle of Lake Tahoe. Like other independent cities in the United States, it is treated as a county-equivalent for census purposes.

History
The Washoe people have inhabited the valley and surrounding areas for about 6,000 years.

The first European Americans to arrive in what is now known as Eagle Valley were John C. Frémont and his exploration party in January 1843. Fremont named the river flowing through the valley Carson River in honor of Kit Carson, the mountain man and scout he had hired for his expedition. Later, settlers named the area Washoe, in reference to the indigenous people.

By 1851, the Eagle Station ranch along the Carson River was a trading post and stopover for travelers on the California Trail's Carson Branch, which ran through Eagle Valley. The valley and trading post received their name from a bald eagle that was hunted and killed by one of the early settlers and was featured on a wall inside the post.

As the area was part of the Utah Territory, it was governed from Salt Lake City, where the territorial government was headquartered. Early settlers bristled at the control by Mormon-influenced officials and desired the creation of the Nevada territory. A vigilante group of influential settlers, headed by Abraham Curry, sought a site for a capital city for the envisioned territory. In 1858, Abraham Curry bought Eagle Station and the settlement was thereafter renamed Carson City. Curry and several other partners had Eagle Valley surveyed for development. Curry decided Carson City would someday serve as the capital city and left a 10 acre plot in the center of town for a capitol building.

After gold and silver were discovered in 1859 on nearby Comstock Lode, Carson City's population began to grow. Curry built the Warm Springs Hotel a mile to the east of the city center. When territorial governor James W. Nye traveled to Nevada, he chose Carson City as the territorial capital, influenced by Carson City lawyer William Stewart, who escorted him from San Francisco to Nevada. As such, Carson City bested Virginia City and American Flat. Curry loaned the Warm Springs Hotel to the territorial Legislature as a meeting hall. The Legislature named Carson City to be the seat of Ormsby County and selected the hotel as the territorial prison, with Curry serving as its first warden. Today, the property is still part of the state prison.

When Nevada became a state in 1864 during the American Civil War, Carson City was confirmed as Nevada's permanent capital. Carson City's development was no longer dependent on the mining industry and instead became a thriving commercial center. The Virginia and Truckee Railroad was built between Virginia City and Carson City. A log flume was also built from the Sierra Nevada into Carson City. The current capitol building was constructed from 1870 to 1871. The United States Mint operated the Carson City Mint between the years 1870 and 1893, which struck gold and silver coins. People came from China during that time, many to work on the railroad. Some of them owned businesses and taught school. By 1880, almost a thousand Chinese people, "one for every five Caucasians", lived in Carson City.

Carson City's population and transportation traffic decreased when the Central Pacific Railroad built a branch line through Donner Pass to connect with the Carson and Colorado Railroad. The new bypassed the Virginia & Truckee, and ran too far to the north to benefit Carson City. The city was slightly revitalized with the mining booms in Tonopah and Goldfield. The US federal building (now renamed the Paul Laxalt Building) was completed in 1890 as was the Stewart Indian School. Even these developments could not prevent its population from dropping to just over 1,500 people by 1930. Carson City resigned itself to small city status, advertising itself as "America's smallest capital". The city slowly grew after World War II; by 1960, it had reached its 1880 boom-time population.

20th-century revitalization and growth
As early as the late 1940s, discussions began about merging Ormsby County and Carson City. By this time, the county was little more than Carson City and a few hamlets to the west. By the 1960 census, all but 2,900 of the county's residents lived in Carson City. However, the effort did not pay off until 1966, when a statewide referendum approved the merger. The required constitutional amendment was passed in 1968. On April 1, 1969, Ormsby County and Carson City officially merged as the Consolidated Municipality of Carson City. With this consolidation, Carson City absorbed former town sites such as Empire City, which had grown up in the 1860s as a milling center along the Carson River and current U.S. Route 50. Carson City could now advertise itself as one of America's largest state capitals with its 146 sqmi of city limits.

In 1991, the city adopted a downtown master plan, specifying no building within 500 ft of the capitol would surpass it in height. This plan effectively prohibited future high-rise development in the center of downtown. The Ormsby House is the tallest building in downtown Carson City, at a height of 117 feet. The structure was completed in 1972.

Geography
Most of the city proper resides in the Eagle Valley. The Carson River flows from Douglas County through the southwestern edge of both the valley and Carson City. Since the consolidation, the city limits today include several small populated areas outside of this valley. Today the city limits include several peaks in the Sierra Nevada, small portions of both the Virginia Range and the Pine Nut Mountains and portions of Marlette Lake and Lake Tahoe. The highest elevation in city limits is Snow Valley Peak at an elevation of 9214 ft. Carson City is one of two state capitals that border another state, the other being Trenton, New Jersey.

Climate
Carson City features a cold semi-arid climate (Koppen: BSk) with cold winters and hot summers. The city is in a high desert river valley approximately 4802 ft above sea level. There are four fairly distinct seasons. Winters see typically light to moderate snowfall, with an average of 14.0 in, with the most snowfall being 82.1 in from July 1951 to June 1952 and the least 3.1 in from July 2002 to June 2003. Most precipitation occurs in winter and spring, with summer and fall being fairly dry, drier than neighboring California. The wettest “rain year” was from July 1937 to June 1938 with 19.36 in and the driest from July 1971 to June 1972 with 3.48 in. The most precipitation in one month occurred in December 1955 when 10.39 in fell and the most snowfall 34.5 in in March 1952. The most precipitation in one day has been 3.12 in on November 19 of 1950.

There are 39.5 afternoons of 90 °F+ highs annually, with 100 °F+ temperatures occurring 1.2 afternoons per year. The hottest month has been July 2021 with an average of 77.6 F, the hottest temperature 107 F on July 19, 1931, and the highest minimum 75 F on August 1, 2022.

There are 125 mornings with lows below freezing, but afternoon maxima top 50 F on all but 52 days, and top freezing on all but five. Temperatures below 0 F are very rare, occurring about twice per winter and frequently not occurring at all. The coldest temperature in Carson City has been −27 F on January 21, 1937, the lowest maximum 5 F on December 12, 1932 and December 22, 1990, and the coldest month January 1949 with a mean temperature of 12.6 F, although January 1937 at 14.9 F is the only other month below 21 F.

The average temperature in Carson City increased by 4.1 F-change between 1984 and 2014, a greater change than in any other city in the United States.

Museums

 * Nevada State Capitol – original capitol still housing the governor's offices with museum exhibits
 * Nevada State Museum – former branch of the United States Mint featuring rock, mining and prehistoric exhibits, and a recreated Wild West village
 * Nevada State Railroad Museum – featuring the Inyo locomotive and relocated Wabuska Railroad Station
 * Stewart Indian School – museum collection includes items from former faculty, students and school
 * Foreman-Roberts House Museum – Gothic Revival architecture, tours available.
 * Sears–Ferris House (not open to public) – home of George Washington Gale Ferris Jr., inventor of the Ferris wheel
 * Yesterday's Flyers, an aviation museum in Carson City.
 * Children's Museum of Northern Nevada – Carson City

Open land

 * Silver Saddle Ranch
 * Mexican Dam – 1860s stone dam across the Carson River
 * Prison Hill – California Trail historic markers, location of the Stewart "S"
 * Carson Aquatic Trail
 * Humboldt-Toiyabe National Forest (Carson Ranger District)
 * Kings Canyon Falls
 * Snow Valley Peak – 9214 ft – highest point within Carson City
 * Tahoe Rim Trail
 * Lake Tahoe–Nevada State Park
 * Lake Tahoe beachfront (several beaches along Lake Tahoe lie within the city limits)
 * Chimney Beach
 * Secret Harbor
 * Whale Beach
 * Skunk Harbor
 * Washoe Lake State Park – borders city to the north
 * "C Hill" – hill featuring the Carson City "C" and giant American Flag

Demographics
Carson City is the smallest of the United States' 366 metropolitan statistical areas.

As of the 2010 census, there were 55,274 people, 20,171 households, and 13,252 families residing in the city. The population density was 366 /mi2. There were 21,283 housing units at an average density of 148 /mi2. The racial makeup of the city was 81.1% White, 1.9% Black or African American, 2.4% Native American, 2.1% Asian, 0.2% Pacific Islander, 9.4% from other races, and 2.9% from two or more races. 21% of the population were Hispanic or Latino of any race.

As of the 2000 census, there were 20,171 households, out of which 29.8% had children under the age of 18 living with them, 50.0% were married couples living together, 11.0% had a female householder with no husband present, and 34.3% were non-families. 27.8% of all households were made up of individuals, and 11.00% had someone living alone who was 65 years of age or older. The average household size was 2.44 and the average family size was 2.97. The city's age distribution was: 23.4% under the age of 18, 7.9% from 18 to 24, 28.9% from 25 to 44, 24.9% from 45 to 64, and 14.9% who were 65 years of age or older. The median age was 39 years. For every 100 females, there were 106.9 males. For every 100 females age 18 and over, there were 108.2 males.

Data from the 2000 census indicates the median income for a household in the city was $41,809, and the median income for a family was $49,570. Males had a median income of $35,296 versus $27,418 for females. The per capita income for the city was $20,943. 10.0% of the population and 6.9% of families were below the poverty line. Out of the total population, 13.7% of those under the age of 18 and 5.8% of those 65 and older were living below the poverty line.

Languages
As of 2010, 82.3% (42,697) of Carson City residents age 5 and older spoke English at home as a first language, while 14.1% (7,325) spoke Spanish, 0.6% (318) French, and numerous Indo-Aryan languages were spoken as a main language by 0.5% (261) of the population over the age of five. In total, 17.7% (9,174) of Carson City's population age 5 and older spoke a first language other than English.

Government and politics
Ormsby County consolidated with Carson City in 1969, and the county simultaneously dissolved. The city is now governed by a five-member board of supervisors, consisting of a mayor and four supervisors. All members are elected at-large, but each of the four supervisors must reside in respective wards, numbered 1 through 4. The mayor and supervisors serve four year terms. Elections are staggered so the mayor and the supervisors from Wards 2 and Ward 4 are elected in presidential election years, and the supervisors from Wards 1 and 3 are elected in the even-numbered years in between (i.e., the same year as gubernatorial elections).

The city is generally considered a Republican stronghold, often voting for Republicans by wide margins. In 2004, George W. Bush defeated John Kerry 57–40%. In 2008, however, Barack Obama became the first Democrat since 1964 to win Ormsby County/Carson City, defeating John McCain 49–48%, by 204 votes, a margin of under 1%.

Carson City, being the state capital, has seen many political protests and demonstrations.

In an attempt to either make a proposed spent nuclear fuel storage facility at Yucca Mountain prohibitively expensive (by raising property tax rates to the maximum allowed) or to allow the state to collect the potential federal payments of property taxes on the facility, the state government in 1987 carved Yucca Mountain out of Nye County and created a new county with no residents out of the area surrounding Yucca called Bullfrog County. Carson City became the county seat of Bullfrog County, even though it is not in Bullfrog County and is more than 100 mi from Yucca Mountain. A state judge found the process unconstitutional in 1989, and Bullfrog County's territory was retroceded to Nye County.

Sports and recreation
Carson City has never hosted any professional team sports. However, a variety of sports are offered at parks and recreation. Many neighborhood parks offer a wide variety of features including picnic tables, beaches, restrooms, fishing, softball, basketball hoops, ponds, tennis, and volleyball. The largest park is Mills Park, which has a total land area of 51 acre and includes the narrow-gauge Carson & Mills Park Railroad. While there are no ski slopes within Carson City, the city is near the Heavenly Mountain Resort, Diamond Peak and Mount Rose Ski Tahoe skiing areas.

Notable people
Carson City has served as one of the state's centers for politics and business. Every state governor since Denver S. Dickerson has resided in the Governor's Mansion in Carson City. The following personalities took up residence in Carson City at some point in their lives.
 * Mackena Bell, racing driver
 * Duane Leroy Bliss, timber businessman
 * Orion Clemens, Secretary of Nevada Territory
 * Steven S. Coughlin, American epidemiologist and author
 * John Cradlebaugh, first Delegate to the U.S. House of Representatives from Nevada Territory
 * Abraham Curry, founding father of Carson City and early politician
 * Dat So La Lee, Native American basket weaver and artist
 * Nellie Verrill Mighels Davis, journalist
 * David Eddings, best selling author of fantasy novels
 * George Washington Gale Ferris Jr., inventor of the Ferris wheel
 * Ellen Hopkins, author
 * Paul Laxalt, former Governor and U.S. Senator
 * Greg LeMond, two time World Champion road racing cyclist, and three-time winner of the Tour de France
 * Alice Little, Irish-American sex-worker and advocate
 * David Lundquist, Major League baseball player (Chicago White Sox)
 * Maurice E. McLoughlin, two-time U.S. Open champion, member of International Tennis Hall of Fame
 * Henry Rust Mighels, journalist, politician, first husband of Nellie Verrill Mighels Davis
 * Hank Monk, stagecoach driver
 * William Ormsby, soldier and namesake of Ormsby County and Ormsby House
 * Donovan Osborne, Major League baseball player (St. Louis Cardinals)
 * Darrell Rasner, Major League baseball player (New York Yankees)
 * Don Tatro, member of the Nevada Senate
 * Mark Twain, author (lived with his brother Orion Clemens)
 * Matt Williams, Major League third baseman (San Francisco Giants, Cleveland Indians, and Arizona Diamondbacks)
 * Sarah Winnemucca, Native American activist and author

Economy and infrastructure
The following is a list of notable employers in Carson City from the fourth quarter of 2012:

1,000–1,499 employees 500–999 employees 200–499 employees 100–199 employees
 * Carson City School District
 * Nevada Department of Transportation
 * Western Nevada College
 * Nevada Department of Corrections
 * Nevada Department of Motor Vehicles
 * Casino Fandango
 * Walmart
 * Precision Castparts Corp.
 * Gold Dust West Hotel and Casino
 * Carson Nugget
 * Costco Wholesale Corporation
 * Nevada Department of Conservation and Natural Resources
 * Nevada Department of Health and Human Services, Division of Welfare and Supportive Services

Transportation
There are four highways in the city: Nevada State Route 28, U.S. Route 395, U.S. Route 50, and Interstate 580, its only freeway. Phase 1 of the Carson City Freeway Project from US 395, just north of the city, to US 50 was completed in February 2006, and Phase 2A, extending from Rt. 50 to Fairview Drive, was officially opened on September 24, 2009. Phase 2B, Fairview Drive to Rt. 50, was completed in August 2017. Prior to 2012, Carson City was one of only five state capitals not directly served by an interstate highway; the city lost this distinction when I-580 was extended into the city limits.

Carson City's first modern bus system, Jump Around Carson, or JAC, opened to the public in October 2005. JAC uses a smaller urban bus ideal for Carson City. Tahoe Transportation District connects Gardnerville with Carson City.

However, there is virtually no ground public transportation to other destinations. Passenger trains have not served Carson City since 1950, when the Virginia and Truckee Railroad was shut down. Greyhound Lines stopped their bus services to the town in 2006 and Amtrak discontinued their connecting thruway bus to Sacramento, California, in 2008. There is now only a limited Monday – Friday RTC bus service, to Reno which is still served by both Greyhound and Amtrak, as well as Eastern Sierra Transit Authority service from Lone Pine to Reno.

Carson City is also served by the Carson Airport, which is a regional airport in the northern part of the city. Reno–Tahoe International Airport, which is 28 mi away, handles domestic commercial flights.

Education
The Carson City School District, the sole public school district of the city, operates ten schools there. The six elementary schools are Bordewich-Bray Elementary School, Empire Elementary School, Fremont Elementary School, Fritsch Elementary School, Mark Twain Elementary School, and Al Seeliger Elementary School. The two middle schools are Carson Middle School and Eagle Valley Middle School. Carson High School and the alternative Pioneer High School serve high school students. Carson High is on Saliman Road.

The district sponsors Carson Montessori School, a public charter school serving grades K-6. Students residing in any Nevada county may enroll. Carson Montessori School is the only school in district operating with a balanced budget. In 2019 Carson Montessori School received the Governor's STEM Schools Designation, an official recognition given to 25 schools statewide which causes a short ceremony attended by the governor during which receiving schools are assigned a 10-foot banner.

Western Nevada College (WNC) is a regionally accredited, two-year and four-year institution which is part of the Nevada System of Higher Education. The college offers many programs including education, arts and science.

Carson City has a public library, the Carson City Library.

title: Carson City, Nevada


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Classification of finite simple groups

In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.

Statement of the classification theorem

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by after Aschbacher and Smith published a 1221-page proof for the missing quasithin case.

Overview of the proof of the classification theorem
wrote two volumes outlining the low rank and odd characteristic part of the proof, and wrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows:

Groups of small 2-rank
The simple groups of low 2-rank are mostly groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.

The simple groups of small 2-rank include: The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.
 * Groups of 2-rank 0, in other words groups of odd order, which are all solvable by the Feit–Thompson theorem.
 * Groups of 2-rank 1. The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or generalized quaternion, which are handled with the Brauer–Suzuki theorem: in particular there are no simple groups of 2-rank 1 except for the cyclic group of order two.
 * Groups of 2-rank 2. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U3(4). The first case was done by the Gorenstein–Walter theorem which showed that the only simple groups are isomorphic to L2(q) for q odd or A7, the second and third cases were done by the Alperin–Brauer–Gorenstein theorem which implies that the only simple groups are isomorphic to L3(q) or U3(q) for q odd or M11, and the last case was done by Lyons who showed that U3(4) is the only simple possibility.
 * Groups of sectional 2-rank at most 4, classified by the Gorenstein–Harada theorem.

All groups not of small 2 rank can be split into two major classes: groups of component type and groups of characteristic 2 type. This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type. (For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.)

Groups of component type
A group is said to be of component type if for some centralizer C of an involution, C/O(C) has a component (where O(C) is the core of C, the maximal normal subgroup of odd order). These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups. A major step in this case is to eliminate the obstruction of the core of an involution. This is accomplished by the B-theorem, which states that every component of C/O(C) is the image of a component of C.

The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimple group, which can be assumed to be already known by induction. So to classify these groups one takes every central extension of every known finite simple group, and finds all simple groups with a centralizer of involution with this as a component. This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.

Groups of characteristic 2 type
A group is of characteristic 2 type if the generalized Fitting subgroup F*(Y) of every 2-local subgroup Y is a 2-group. As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

The rank 1 groups are the thin groups, classified by Aschbacher, and the rank 2 ones are the notorious quasithin groups, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.

Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4. The three classes are groups of GF(2) type (classified mainly by Timmesfeld), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem and work by several others), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups. The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.

Existence and uniqueness of the simple groups
The main part of the classification produces a characterization of each simple group. It is then necessary to check that there exists a simple group for each characterization and that it is unique. This gives a large number of separate problems; for example, the original proofs of existence and uniqueness of the monster group totaled about 200 pages, and the identification of the Ree groups by Thompson and Bombieri was one of the hardest parts of the classification. Many of the existence proofs and some of the uniqueness proofs for the sporadic groups originally used computer calculations, most of which have since been replaced by shorter hand proofs.

Gorenstein's program
In 1972 announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:
 * 1) Groups of low 2-rank. This was essentially done by Gorenstein and Harada, who classified the groups with sectional 2-rank at most 4. Most of the cases of 2-rank at most 2 had been done by the time Gorenstein announced his program.
 * 2) The semisimplicity of 2-layers. The problem is to prove that the 2-layer of the centralizer of an involution in a simple group is semisimple.
 * 3) Standard form in odd characteristic. If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in "standard form" meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.
 * 4) Classification of groups of odd type. The problem is to show that if a group has a centralizer of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution theorem.
 * 5) Quasi-standard form
 * 6) Central involutions
 * 7) Classification of alternating groups.
 * 8) Some sporadic groups
 * 9) Thin groups. The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified by Aschbacher in 1978
 * 10) Groups with a strongly p-embedded subgroup for p odd
 * 11) The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors. This was solved by McBride in 1982.
 * 12) Groups of characteristic p type. This is the problem of groups with a strongly p-embedded 2-local subgroup with p odd, which was handled by Aschbacher.
 * 13) Quasithin groups. A quasithin group is one whose 2-local subgroups have p-rank at most 2 for all odd primes p, and the problem is to classify the simple ones of characteristic 2 type. This was completed by Aschbacher and Smith in 2004.
 * 14) Groups of low 2-local 3-rank. This was essentially solved by Aschbacher's trichotomy theorem for groups with e(G)=3. The main change is that 2-local 3-rank is replaced by 2-local p-rank for odd primes.
 * 15) Centralizers of 3-elements in standard form. This was essentially done by the Trichotomy theorem.
 * 16) Classification of simple groups of characteristic 2 type. This was handled by the Gilman–Griess theorem, with 3-elements replaced by p-elements for odd primes.

Timeline of the proof
Many of the items in the table below are taken from. The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the "wrong" order.

Second-generation classification
The proof of the theorem, as it stood around 1985 or so, can be called first generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called a second-generation classification proof. This effort, called "revisionism", was originally led by Daniel Gorenstein.

, ten volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005, 2018a, 2018b; & Capdeboscq, 2021, 2023). In 2012 Solomon estimated that the project would need another 5 volumes, but said that progress on them was slow. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from the second generation proof being written in a more relaxed style.) However, with the publication of volume 9 of the GLS series, and including the Aschbacher–Smith contribution, this estimate was already reached, with several more volumes still in preparation (the rest of what was originally intended for volume 9, plus projected volumes 10 and 11). Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof.

Gorenstein and his collaborators have given several reasons why a simpler proof is possible.
 * The most important thing is that the correct, final statement of the theorem is now known. Simpler techniques can be applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those who worked on the first generation proof did not know how many sporadic groups there were, and in fact some of the sporadic groups (e.g., the Janko groups) were discovered while proving other cases of the classification theorem. As a result, many of the pieces of the theorem were proved using techniques that were overly general.
 * Because the conclusion was unknown, the first generation proof consists of many stand-alone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of numerous special cases. Given a larger, orchestrated proof, dealing with many of these special cases can be postponed until the most powerful assumptions can be applied. The price paid under this revised strategy is that these first generation theorems no longer have comparatively short proofs, but instead rely on the complete classification.
 * Many first generation theorems overlap, and so divide the possible cases in inefficient ways. As a result, families and subfamilies of finite simple groups were identified multiple times. The revised proof eliminates these redundancies by relying on a different subdivision of cases.
 * Finite group theorists have more experience at this sort of exercise, and have new techniques at their disposal.

has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, a third generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.

Length of proof
Gorenstein has discussed some of the reasons why there might not be a short proof of the classification similar to the classification of compact Lie groups.

 * The most obvious reason is that the list of simple groups is quite complicated: with 26 sporadic groups there are likely to be many special cases that have to be considered in any proof. So far no one has yet found a clean uniform description of the finite simple groups similar to the parameterization of the compact Lie groups by Dynkin diagrams.
 * Atiyah and others have suggested that the classification ought to be simplified by constructing some geometric object that the groups act on and then classifying these geometric structures. The problem is that no one has been able to suggest an easy way to find such a geometric structure associated with a simple group. In some sense, the classification does work by finding geometric structures such as BN-pairs, but this only comes at the end of a very long and difficult analysis of the structure of a finite simple group.
 * Another suggestion for simplifying the proof is to make greater use of representation theory. The problem here is that representation theory seems to require very tight control over the subgroups of a group in order to work well. For groups of small rank, one has such control and representation theory works very well, but for groups of larger rank no-one has succeeded in using it to simplify the classification. In the early days of the classification, there was a considerable effort made to use representation theory, but this never achieved much success in the higher rank case.

Consequences of the classification
This section lists some results that have been proved using the classification of finite simple groups.

 * The Schreier conjecture
 * The Signalizer functor theorem
 * The B conjecture
 * The Schur–Zassenhaus theorem for all groups (though this only uses the Feit–Thompson theorem).
 * A transitive permutation group on a finite set with more than 1 element has a fixed-point-free element of prime power order.
 * The classification of 2-transitive permutation groups.
 * The classification of rank 3 permutation groups.
 * The Sims conjecture
 * Frobenius's conjecture on the number of solutions of xn = 1.

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

title: Classification of finite simple groups


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Chalcolithic

The Chalcolithic (also called the Copper Age and Eneolithic) was an archaeological period characterized by the increasing use of smelted copper. It followed the Neolithic and preceded the Bronze Age. It occurred at different periods in different areas, but was absent in some parts of the world, such as Russia, where there was no well-defined Copper Age between the Stone and Bronze ages. Stone tools were still predominantly used during this period.

The archaeological site of Belovode, on Rudnik mountain in Serbia, has the world's oldest securely dated evidence of copper smelting at high temperature, from c. 5,000 BC. The transition from Copper Age to Bronze Age in Europe occurred between the late 5th and the late 3rd millennium BC. In the Ancient Near East the Copper Age covered about the same period, beginning in the late 5th millennium BC and lasting for about a millennium before it gave rise to the Early Bronze Age.

A study in the journal Antiquity from 2013 reporting the discovery of a tin bronze foil from the Pločnik archaeological site dated to c. 4,650 BC, as well as 14 other artefacts from Bulgaria and Serbia dated to before 4,000 BC, showed that early tin bronze was more common than previously thought and developed independently in Europe 1,500 years before the first tin bronze alloys in the Near East. In Britain, the Chalcolithic is a short period between about 2,500 and 2,200 BC, characterized by the first appearance of objects of copper and gold, a new ceramic culture and the immigration of Beaker culture people, heralding the end of the local late Neolithic.

Terminology
The multiple names result from multiple definitions of the period. Originally, the term Bronze Age meant that either copper or bronze was being used as the chief hard substance for the manufacture of tools and weapons. Ancient writers, who provided the essential cultural references for educated people during the 19th century, used the same name for both copper- and bronze-using ages.

The concept of the Copper Age was put forward by Hungarian scientist Ferenc Pulszky in the 1870s, when, on the basis of the significant number of large copper objects unearthed within the Carpathian Basin, he suggested that the previous threefold division of the Prehistoric Age – the Stone, Bronze and Iron Ages – should be further divided with the introduction of the Copper Age.

In 1881, John Evans recognized that use of copper often preceded the use of bronze, and distinguished between a transitional Copper Age and the Bronze Age proper. He did not include the transitional period in the Bronze Age, but described it separately from the customary stone / bronze / iron system, at the Bronze Age's beginning. He did not, however, present it as a fourth age but chose to retain the tripartite system.

In 1884, Gaetano Chierici, perhaps following the lead of Evans, renamed it in Italian as the eneo-litica, or "bronze–stone" transition. The phrase was never intended to mean that the period was the only one in which both bronze and stone were used. The Copper Age features the use of copper, excluding bronze; moreover, stone continued to be used throughout both the Bronze Age and the Iron Age. The part -litica simply names the Stone Age as the point from which the transition began and is not another -lithic age.

Subsequently, British scholars used either Evans's "Copper Age" or the term "Eneolithic" (or Æneolithic), a translation of Chierici's eneo-litica. After several years, a number of complaints appeared in the literature that "Eneolithic" seemed to the untrained eye to be produced from e-neolithic, "outside the Neolithic", clearly not a definitive characterization of the Copper Age. Around 1900, many writers began to substitute Chalcolithic for Eneolithic, to avoid the false segmentation. The term chalcolithic is a combination of two words- Chalco+Lithic, derived from the Greek words "khalkos" meaning "copper", and "líthos" meaning "stone".

But "chalcolithic" could also mislead: For readers unfamiliar with the Italian language, chalcolithic seemed to suggest another -lithic age, paradoxically part of the Stone Age despite the use of copper. Today, Copper Age, Eneolithic, and Chalcolithic are used synonymously to mean Evans's original definition of Copper Age.

Near East

The emergence of metallurgy may have occurred first in the Fertile Crescent.

Lead may have been the first ore that humans smelted, since it can be easily obtained by heating galena.

Possible early examples of lead smelting, supported by the extreme rarity of native lead, include: lead beads, found on Level IX of Chatal/Çatal Hüyük in central Anatolia, though they might be made of galena, cerussite, or metallic lead, and accordingly might or might not be evidence of early smelting; a lead bead, found in a GK59 group test square in the 4th level of Jarmo, dated to the 7th millennium BCE, though it is small enough that its human usage is doubtful; a lead bracelet, found in level XII of Yarim Tepe I, dated to the 6th millennium BC; a small cone-shaped piece of lead, found in the "Burnt House" in TT6 at Arpachiyah, dated to the Halaf period or slightly later than the Yarim Tepe bracelet; and more.

Copper smelting is also documented at this site at about the same time period (soon after 6000 BC). However, the use of lead seems to precede copper smelting. Early metallurgy is also documented at the nearby site of Tell Maghzaliyah, which seems to be dated even earlier, and completely lacks pottery.

The Timna Valley contains evidence of copper mining in 7000–5000 BC. The process of transition from Neolithic to Chalcolithic in the Middle East is characterized in archaeological stone tool assemblages by a decline in high quality raw material procurement and use. This dramatic shift is seen throughout the region, including the Tehran Plain, Iran. Here, analysis of six archaeological sites determined a marked downward trend in not only material quality, but also in aesthetic variation in the lithic artefacts. Fazeli & Coningham use these results as evidence of the loss of craft specialisation caused by increased use of copper tools.

The Tehran Plain findings illustrate the effects of the introduction of copper working technologies on the in-place systems of lithic craft specialists and raw materials. Networks of exchange and specialized processing and production that had evolved during the Neolithic seem to have collapsed by the Middle Chalcolithic (c. 4500–3500 BC) and been replaced by the use of local materials by a primarily household-based production of stone tools.

Arsenical copper or bronze was produced in eastern Turkey (Malatya Province) at two ancient sites, Norşuntepe and Değirmentepe, around 4200 BC. According to Boscher (2016), hearths or natural draft furnaces, slag, ore, and pigment had been recovered throughout these sites. This was in the context of Ubaid period architectural complexes typical of southern Mesopotamian architecture. Norşuntepe site demonstrates that some form of arsenic alloying was indeed taking place by the 4th millennium BC. Since the slag identified at Norşuntepe contains no arsenic, this means that arsenic in some form was added separately.

Europe
A copper axe found at Prokuplje, Serbia contains the oldest securely dated evidence of copper-making, c. 5500 BC (7,500 years ago). The find in June 2010 extends the known record of copper smelting by about 800 years, and suggests that copper smelting may have been invented in separate parts of Asia and Europe at that time rather than spreading from a single source.

Knowledge of the use of copper was far more widespread than the metal itself. The European Battle Axe culture used stone axes modeled on copper axes, even with moulding carved in the stone. Ötzi the Iceman, who was found in the Ötztal Alps in 1991 and whose remains have been dated to about 3300 BC, was found with a Mondsee copper axe.

Examples of Chalcolithic cultures in Europe include Vila Nova de São Pedro and Los Millares on the Iberian Peninsula. Pottery of the Beaker people has been found at both sites, dating to several centuries after copper-working began there. The Beaker culture appears to have spread copper and bronze technologies in Europe, along with Indo-European languages. In Britain, copper was used between the 25th and 22nd centuries BC, but some archaeologists do not recognise a British Chalcolithic because production and use was on a small scale.

South Asia
Ceramic similarities between the Indus Valley civilisation, southern Turkmenistan, and northern Iran during 4300–3300 BC of the Chalcolithic period suggest considerable mobility and trade.

The term "Chalcolithic" has also been used in the context of the South Asian Stone Age.

In Bhirrana, the earliest Indus civilization site, copper bangles and arrowheads were found. The inhabitants of Mehrgarh in present-day Pakistan fashioned tools with local copper ore between 7000 and 3300 BC.

The Nausharo site was a pottery workshop in province of Balochistan, Pakistan, that dates to 4,500 years ago; 12 blades and blade fragments were excavated there. These blades are 12 - 18 cm long, 1.2 - 2.0 cm wide, and relatively thin. Archaeological experiments show that these blades were made with a copper indenter and functioned as a potter's tool to trim and shape unfired pottery. Petrographic analysis indicates local pottery manufacturing, but also reveals the existence of a few exotic black-slipped pottery items from the Indus Valley.

In India, Chalcolithic culture flourished in mainly four farming communities – Ahar or Banas, Kayatha, Malwa, and Jorwe. These communities had some common traits like painted pottery and use of copper, but they had a distinct ceramic design tradition. Banas culture (2000–1600 BC) had ceramics with red, white, and black design. Kayatha culture (2450–1700 BC) had ceramics painted with brown colored design. Malwa culture (1900–1400 BC) had profusely decorated pottery with red or black colored design. Jorwe culture (1500–900 BC) had ceramics with matte surface and black-on-red design.

Pandu Rajar Dhibi (2000–1600 BC) is a Chalcolithic site in the eastern part of the Indian subcontinent. It is located on the south bank of Ajay River in West Bengal. Blackware, painted Koshi ware, pottery, various ornaments made of pearl and copper, various types of tools, pieces of fabric woven from Shimul cotton thread, human and various animal skeletons, burnt clay fragments have been found at the site.

In March 2018, archaeologists had discovered three carts and copper artifacts including weapons dating to 1800 BC in Sanauli village of Uttar Pradesh. The artifacts belongs to Ochre Coloured Pottery culture.

Pre-Columbian Americas
Andean civilizations in South America appear to have independently invented copper smelting.

The term "Chalcolithic" is also applied to American civilizations that already used copper and copper alloys thousands of years before Europeans immigrated. Besides cultures in the Andes and Mesoamerica, the Old Copper complex mined and fabricated copper as tools, weapons, and personal ornaments in an area centered in the upper Great Lakes region: Present-day Michigan and Wisconsin.

The evidence of smelting or alloying that has been found in North America is subject to some dispute and a common assumption by archaeologists is that objects were cold-worked into shape. Artifacts from some of these sites have been dated to 6500–1000 BC, making them some of the oldest Chalcolithic sites in the world. Some archaeologists find artifactual and structural evidence of casting by Hopewellian and Mississippian peoples to be demonstrated in the archaeological record.

East Asia
In the 5th millennium BC copper artifacts start to appear in East Asia, such as in the Jiangzhai and Hongshan cultures, but those metal artifacts were not widely used during this early stage.

Copper manufacturing gradually appeared in the Yangshao period (5000–3000 BC). Jiangzhai is the only site where copper artifacts were found in the Banpo culture. Archaeologists have found remains of copper metallurgy in various cultures from the late fourth to the early third millennia BC. These include the copper-smelting remains and copper artifacts of the Hongshan culture (4700–2900) and copper slag at the Yuanwozhen site. This indicates that inhabitants of the Yellow River valley had already learned how to make copper artifacts by the later Yangshao period.

Sub-Saharan Africa
In the region of the Aïr Mountains, Niger, independent copper smelting developed between 3000 and 2500 BC. The process was not in a developed state, indicating smelting was not foreign. It became mature about 1500 BC.

title: Chalcolithic