# 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

## 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,**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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