Higman group

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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 Gn,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


  • 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, 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, ISBN 978-0-7081-0300-5, MR 0376874