Higman group

In mathematics, the Higman group, introduced by Graham Higman (1951), 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. Higman (1974) later found some finitely presented infinite groups $G n, r$ that are simple if $n$ is even and have a simple subgroup of index 2 if $n$ is odd, one of which is one of the Thompson groups.

Higman's group is generated by 4 elements $a, b, c, d$ with the relations