Higman group explained

In mathematics, the Higman group, introduced by, was the first example of an infinite finitely presented group with no nontrivial 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

a ^{-1} ba = b ^{2,} b ^{-1} cb = c ^{2,} c ^{-1} dc = d ^{2,} d ^{-1} ad = a ^{2.}