Editing Simple group (section)

=== 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>