Editing Simple group

{{Short description|Group without normal subgroups other than the trivial group and itself}}
{{Group theory sidebar |Basics}}

In [[mathematics]], a '''simple group''' is a nontrivial [[Group (mathematics)|group]] whose only [[normal subgroup]]s are the [[trivial group]] and the group itself.  A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding [[quotient group]]. This process can be repeated, and for [[finite group]]s one eventually arrives at uniquely determined simple groups, by the [[Jordan–Hölder theorem]].

The complete [[classification of finite simple groups]], completed in 2004, is a major milestone in the history of mathematics.

== Examples ==

=== Finite simple groups ===
The [[cyclic group]] <math>G=(\mathbb{Z}/3\mathbb{Z},+)=\mathbb{Z}_3</math> of [[congruence class]]es [[Modulo operation|modulo]] 3 (see  [[modular arithmetic]]) is simple. If <math>H</math> is a subgroup of this group, its [[Order (group theory)|order]] (the number of elements) must be a [[divisor]] of the order of <math>G</math> which is 3. Since 3 is prime, its only divisors are 1 and 3, so either <math>H</math> is <math>G</math>, or <math>H</math> is the trivial group. On the other hand, the group <math>G=(\mathbb{Z}/12\mathbb{Z},+)=\mathbb{Z}_{12}</math> is not simple. The set <math>H</math> of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an [[abelian group]] is normal. Similarly, the additive group of the [[integer]]s <math>(\mathbb{Z},+)</math> is not simple; the set of even integers is a non-trivial proper normal subgroup.<ref>Knapp (2006), [{{Google books|plainurl=y|id=KVeXG163BggC|page=170|text=Z is not simple, having the nontrivial subgroup 2Z}} p. 170]</ref>

One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of [[prime number|prime]] order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the [[alternating group]] <math>A_5</math> of order 60, and every simple group of order 60 is [[Group isomorphism|isomorphic]] to <math>A_5</math>.<ref>Rotman (1995), [{{Google books|plainurl=y|id=lYrsiaHSHKcC|page=226|text=simple groups of order 60 are isomorphic}} p. 226]</ref> The second smallest nonabelian simple group is the projective special linear group [[PSL(2,7)]] of order 168, and every simple group of order 168 is isomorphic to PSL(2,7).<ref>Rotman (1995), p. 281</ref><ref>Smith & Tabachnikova (2000), [{{Google books|plainurl=y|id=DD0TW28WjfQC|page=144|text=any two simple groups of order 168 are isomorphic}} p. 144]</ref>

=== Infinite simple groups ===
The infinite alternating group <math>A_\infty</math>, i.e. the group of even finitely supported permutations of the integers, is simple. This group can be written as the increasing union of the finite simple groups <math>A_n</math> with respect to standard embeddings <math>A_n \rightarrow A_{n+1}</math>. Another family of examples of infinite simple groups is given by <math>PSL_n(F)</math>, where <math>F</math> is an infinite field and <math>n\geq2</math>.

It is much more difficult to construct ''finitely generated'' infinite simple groups. The first existence result is non-explicit; it is due to [[Graham Higman]] and consists of simple quotients of the [[Higman group]].<ref>{{Citation | last1=Higman | first1=Graham | author1-link=Graham Higman | title=A finitely generated infinite simple group | doi=10.1112/jlms/s1-26.1.59  |mr=0038348 | year=1951 | journal=Journal of the London Mathematical Society |series=Second Series | issn=0024-6107 | volume=26 | issue=1 | pages=61–64}}</ref> Explicit examples, which turn out to be finitely presented, include the infinite [[Thompson groups]] <math>T</math> and <math>V</math>. Finitely presented [[Torsion (algebra)|torsion-free]] infinite simple groups were constructed by Burger and Mozes.<ref>{{cite journal | last1 = Burger | first1 = M. | last2 = Mozes | first2 = S. | year = 2000 | title = Lattices in product of trees | journal = Publ. Math. IHÉS | volume = 92 | pages = 151–194 | doi=10.1007/bf02698916| s2cid = 55003601 }}</ref>

== Classification ==
There is as yet no known classification for general (infinite) simple groups, and no such classification is expected. One reason for this is the existence of continuum-many [[Tarski monster group]]s for every sufficiently-large prime characteristic, each simple and having only the cyclic group of that characteristic as its subgroups.<ref>{{citation|contribution-url=https://www.raczar.es/webracz/ImageServlet?mod=publicaciones&subMod=monografias&car=monografia26&archivo=089Otal.pdf|contribution=The Classification of the Finite Simple Groups: An Overview|first=Javier|last=Otal|title=Problemas del Milenio |editor-first=L. J.|editor-last=Boya|series=Monografías de la Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza|volume=26|publisher=Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza|year=2004}}</ref>

=== Finite simple groups ===
{{main|list of finite simple groups}}
{{further|Classification of finite simple groups}}
The [[List of finite simple groups|finite simple groups]] are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way [[prime number]]s are the basic building blocks of the [[integer]]s. This is expressed by the [[Jordan–Hölder theorem]] which states that any two [[composition series]] of a given group have the same length and the same factors, [[up to]] [[permutation]] and [[isomorphism]]. In a huge collaborative effort, the [[classification of finite simple groups]] was declared accomplished in 1983 by [[Daniel Gorenstein]], though some problems surfaced (specifically in the classification of [[quasithin group]]s, which were plugged in 2004).

Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:
* <math>\mathbb{Z}_p</math> – [[cyclic group]] of prime order
* <math>A_n</math> – [[alternating group]] for <math>n\geq5</math>
*:The alternating groups may be considered as groups of Lie type over the [[field with one element]], which unites this family with the next, and thus all families of non-abelian finite simple groups may be considered to be of Lie type.
* One of 16 families of [[groups of Lie type]] or their derivatives
*:The [[Tits group]] is generally considered of this form, though strictly speaking it is not of Lie type, but rather index 2 in a group of Lie type.
* One of 26 exceptions, the [[sporadic group]]s, of which 20 are subgroups or [[subquotient]]s of the [[monster group]] and are referred to as the "Happy Family", while the remaining 6 are referred to as [[pariah group|pariahs]].

== Structure of finite simple groups ==
The famous [[Feit–Thompson theorem|theorem]] of [[Walter Feit|Feit]] and [[John G. Thompson|Thompson]] states that every group of odd order is [[solvable group|solvable]]. Therefore, every finite simple group has even order unless it is cyclic of prime order.

The [[Schreier conjecture]] asserts that the group of [[outer automorphism]]s of every finite simple group is solvable.  This can be proved using the [[classification theorem]].

== 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.

==Tests for nonsimplicity==
''[[Sylow theorems#Example applications|Sylow's test]]'': Let ''n'' be a positive integer that is not prime, and let ''p'' be a prime divisor of ''n''. If 1 is the only divisor of ''n'' that is congruent to 1 modulo ''p'', then there does not exist a simple group of order ''n''.

Proof: If ''n'' is a prime-power, then a group of order ''n'' has a nontrivial [[center (group theory)|center]]<ref>See the proof in [[p-group|''p''-group]], for instance.</ref> and, therefore, is not simple. If ''n'' is not a prime power, then every Sylow subgroup is proper, and, by [[Sylow theorems|Sylow's Third Theorem]], we know that the number of Sylow ''p''-subgroups of a group of order ''n'' is equal to 1 modulo ''p'' and divides ''n''. Since 1 is the only such number, the Sylow ''p''-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.

''Burnside'': A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from [[Burnside's theorem]].

== See also ==
* [[Almost simple group]]
* [[Characteristically simple group]]
* [[Quasisimple group]]
* [[Semisimple group]]
* [[List of finite simple groups]]

==References==

=== Notes ===
{{reflist}}

=== Textbooks ===
{{refbegin}}
*{{Citation | last1=Wilson | first1=Robert A. | author-link = Robert Arnott Wilson | title=The finite simple groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | zbl=1203.20012 | year=2009 | volume=251}}; [http://www.maths.qmul.ac.uk/~raw/fsgs.html 2007 preprint].
* {{Citation | last1=Burnside | first1=William | author1-link=William Burnside | title=Theory of groups of finite order | publisher=[[Cambridge University Press]] | year=1897}}
{{refend}}
* {{Citation |last1=Knapp |first1=Anthony W. |title=Basic algebra |year=2006 |publisher=Springer |isbn=978-0-8176-3248-9 }}
* {{Citation |last1=Rotman |first1=Joseph J. |title=An introduction to the theory of groups |series=Graduate texts in mathematics |volume=148 |year=1995 |publisher=Springer |isbn=978-0-387-94285-8 }}
* {{Citation |last1=Smith |first1=Geoff |last2=Tabachnikova |first2=Olga |title=Topics in group theory |edition=2 |series=Springer undergraduate mathematics series |year=2000 |publisher=Springer |isbn=978-1-85233-235-8 }}

=== Papers ===
{{refbegin}}
* {{Citation 
| doi = 10.1007/BF00327738 
| title = Simple groups of finite order in the nineteenth century 
| date=September 1979 | last = Silvestri | first = R.
| journal = Archive for History of Exact Sciences 
| volume = 20
| pages = 313–356 
| issue = 3–4
| s2cid = 120444304 
}}
{{refend}}

[[Category:Properties of groups]]