Higman group

 ( Group Theory ) 

In mathematics, the Higman group, introduced by , was the first example of an infinite group finitely presented group with no trivial group finite group quotient group. The quotient by the maximal proper normal subgroup is a finitely generated infinite simple group. later found some finitely presented infinite groups that are simple if is even and have a simple subgroup of index 2 if is odd, one of which is one of the Thompson groups.

Higman's group is generated by 4 elements with the relations

$a^\{-1\}ba\; =\; b^2,\backslash quad\; b^\{-1\}cb\; =\; c^2,\backslash quad\; c^\{-1\}dc\; =\; d^2,\backslash quad\; d^\{-1\}ad\; =\; a^2.$