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

[math]\displaystyle{ a^{-1}ba=b^2,\quad b^{-1}cb=c^2,\quad c^{-1}dc=d^2,\quad d^{-1}ad=a^2. }[/math]

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 
• 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, https://books.google.com/books?id=LPvuAAAAMAAJ