Editing Simple group (section)

== History for finite simple groups ==
There are two threads in the history of finite simple groups – the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the Monster in 1981; and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 (when victory was initially declared), but was only generally agreed to be finished in 2004. By 2018, its publication was envisioned as a series of 12 [[monograph]]s,<ref>{{citation
 | last = Solomon | first = Ronald
 | issue = 6
 | journal = Notices of the American Mathematical Society
 | mr = 3792856
 | pages = 646–651
 | title = The classification of finite simple groups: a progress report
 | url = https://www.ams.org/journals/notices/201806/rnoti-p646.pdf
 | volume = 65
 | year = 2018| doi = 10.1090/noti1689
 }}</ref> the tenth of which was published in 2023.<ref>{{citation
 | last1 = Capdeboscq | first1 = Inna
 | last2 = Gorenstein | first2 = Daniel
 | last3 = Lyons | first3 = Richard
 | last4 = Solomon | first4 = Ronald
 | isbn = 978-1-4704-7553-6
 | mr = 4656413
 | publisher = American Mathematical Society, Providence, RI
 | series = Mathematical Surveys and Monographs
 | title = The classification of the finite simple groups, Number 10. Part V. Chapters 9–17. Theorem <math>C_6</math> and Theorem <math>C^{\ast}_4</math>, Case A
 | volume = 40
 | year = 2023}}</ref> See {{Harv|Silvestri|1979}} for 19th century history of simple groups.

=== Construction ===
Simple groups have been studied at least since early [[Galois theory]], where [[Évariste Galois]] realized that the fact that the [[alternating group]]s on five or more points are simple (and hence not solvable), which he proved in 1831, was the reason that one could not solve the quintic in radicals. Galois also constructed the [[projective special linear group]] of a plane over a prime finite field, {{nowrap|PSL(2,''p'')}}, and remarked that they were simple for ''p'' not 2 or 3. This is contained in his last letter to Chevalier,<ref name="chevalier-letter">{{Citation
 | last = Galois
 | first = Évariste
 | year = 1846
 | title = Lettre de Galois à M. Auguste Chevalier
 | journal = [[Journal de Mathématiques Pures et Appliquées]]
 | volume = XI
 | pages = 408–415
 | url = http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-16390&I=416&M=tdm
 | access-date = 2009-02-04
 | postscript =, PSL(2,''p'') and simplicity discussed on p. 411; exceptional action on 5, 7, or 11 points discussed on pp. 411–412; GL(''ν'',''p'') discussed on p. 410}}</ref> and are the next example of finite simple groups.<ref name="raw">{{citation
|first=Robert
|last=Wilson
|author-link=Robert Arnott Wilson
|date= October 31, 2006 |url=http://www.maths.qmul.ac.uk/~raw/fsgs.html
|title=The finite simple groups
|chapter=Chapter 1: Introduction
|chapter-url=http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps
}}</ref>

The next discoveries were by [[Camille Jordan]] in 1870.<ref>{{citation
|first=Camille
|last=Jordan
|author-link=Camille Jordan
|title=[[List of important publications in mathematics#Trait.C3.A9 des substitutions et des .C3.A9quations alg.C3.A9briques|Traité des substitutions et des équations algébriques]]
|year=1870
}}</ref> Jordan had found 4 families of simple matrix groups over [[finite field]]s of prime order, which are now known as the [[classical group]]s.

At about the same time, it was shown that a family of five groups, called the [[Mathieu group]]s and first described by [[Émile Léonard Mathieu]] in 1861 and 1873, were also simple. Since these five groups were constructed by methods which did not yield infinitely many possibilities, they were called "[[sporadic group|sporadic]]" by [[William Burnside]] in his 1897 textbook.

Later Jordan's results on classical groups were generalized to arbitrary finite fields by [[Leonard Dickson]], following the classification of [[complex simple Lie algebra]]s by [[Wilhelm Killing]]. Dickson also constructed exception groups of type G<sub>2</sub> and [[E6 (mathematics)|E<sub>6</sub>]] as well, but not of types F<sub>4</sub>, E<sub>7</sub>, or E<sub>8</sub> {{harv|Wilson|2009|p=2}}. In the 1950s the work on groups of Lie type was continued, with [[Claude Chevalley]] giving a uniform construction of the classical groups and the groups of exceptional type in a 1955 paper. This omitted certain known groups (the projective unitary groups), which were obtained by "twisting" the Chevalley construction. The remaining groups of Lie type were produced by Steinberg, Tits, and Herzig (who produced <sup>3</sup>''D''<sub>4</sub>(''q'') and <sup>2</sup>''E''<sub>6</sub>(''q'')) and by Suzuki and Ree (the [[Suzuki–Ree group]]s).

These groups (the groups of Lie type, together with the cyclic groups, alternating groups, and the five exceptional Mathieu groups) were believed to be a complete list, but after a lull of almost a century since the work of Mathieu, in 1964 the first [[Janko group]] was discovered, and the remaining 20 sporadic groups were discovered or conjectured in 1965–1975, culminating in 1981, when [[Robert Griess]] announced that he had constructed [[Bernd Fischer (mathematician)|Bernd Fischer]]'s "[[Monster group]]". The Monster is the largest sporadic simple group having order of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. The Monster has a faithful 196,883-dimensional representation in the 196,884-dimensional [[Griess algebra]], meaning that each element of the Monster can be expressed as a 196,883 by 196,883 matrix.

=== Classification ===
The full classification is generally accepted as beginning with the [[Feit–Thompson theorem]] of 1962–1963 and being completed in 2004.

Soon after the construction of the Monster in 1981, a proof, totaling more than 10,000 pages, was supplied in 1983 by Daniel Gorenstein, that claimed to successfully [[List of finite simple groups|list all finite simple groups]]. This was premature, as gaps were later discovered in the classification of [[quasithin group]]s. The gaps were filled in 2004 by a 1300 page classification of quasithin groups and the proof is now generally accepted as complete.