Editing Simple group (section)

== 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>