 Higman group

The Higman group has the finite group presentation $Q=⟨a_1,a_2,a_3,a_4\mid a_i{a}_{i+1}{a}_i^{-1}=a_{i+1}^2(i\mathrm{mod}4)⟩$. Its subgroup $⟨a_1,a_3⟩$ is a rank-two free group, and the normal closure of each $a_i$ is all of $Q$. It has no nontrivial finite quotients of a group: a least prime divisor among the four generator orders in a finite image contradicts the multiplicative-order constraints imposed by conjugation to squares. These properties give finite-presentation constructions embedding arbitrary finitely presented groups into groups with no nontrivial finite quotients.

The Higman group, often denoted as \( \text{H} \), is a notable example of a group in the field of group theory, particularly in the area of infinite groups. It was constructed by Graham Higman in the 1950s as an example of a finitely generated group that is not finitely presented. The Higman group can be defined using a particular way of organizing its generators and relations.