Higman group

For the finite simple group, see Higman–Sims group.

In mathematics, the Higman group, introduced by Graham Higman (1951), 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. 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, doi:10.1112/jlms/s1-26.1.61, ISSN 0024-6107, MR 0038348
• Higman, Graham (1974), Finitely presented infinite simple groups, Notes on Pure Mathematics, vol. 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, ISBN 978-0-7081-0300-5, MR 0376874