information:extraction, structural:scholar_edition, information:classification, stylistic:style_transfer
Infinite simple groups

The infinite alternating group A ∞ {\displaystyle A_{\infty }}, 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 A n {\displaystyle A_{n}} with respect to standard embeddings A n → A n + 1 {\displaystyle A_{n}\rightarrow A_{n+1}}. Another family of examples of infinite simple groups is given by P S L n (F) {\displaystyle PSL_{n}(F)}, where F {\displaystyle F} is an infinite field and n ≥ 2 {\displaystyle n\geq 2}.

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. Explicit examples, which turn out to be finitely presented, include the infinite Thompson groups T {\displaystyle T} and V {\displaystyle V}. Finitely presented torsion-free infinite simple groups were constructed by Burger and Mozes.