Higman group

In mathematics, the Higman group, introduced by , was the first example of an infinite finitely presented group with no non-trivial finite quotients. 
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