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

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

References

Higman, Graham (1951), "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series, 26 (1): 61–64, ISSN 0024-6107, MR 0038348, doi:10.1112/jlms/s1-26.1.61
Higman, Graham (1974), Finitely presented infinite simple groups, Notes on Pure Mathematics, 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, ISBN 978-0-7081-0300-5, MR 0376874

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