---
tag: QGW0Z23XRK
kind: concept
name: Higman group
aliases: [higmans-group]
sources:
  - wikipedia: en:34581441@r1373995312
    locator: Higman group
    note: adopted
    mention: wikipedia://en/34581441
    sections: [Summary]
same_as:
  wikidata: Q5760134
  wikipedia: Higman group
inventory: [1M1NKDAVZH]
---

## Summary

In mathematics, the **Higman group**, introduced by (txt), was the first example of an infinite finitely presented group with no [[nontrivial|0BNPC9850Y]] finite [[quotients|0KYX2N69J0]]. The quotient by the maximal proper [[normal subgroup|0CB0DX5BVV]] is a [[finitely generated|082DWWXMRJ]] infinite [[simple group|0C093D1T4M]]. (Higman 1974) later found some finitely presented infinite groups $G_{n,r}$ that are simple if $n$ is even and have a simple [[subgroup|0X1JXT47GA]] of [[index|0DTXBQPA2S]] 2 if $n$ is odd, one of which is one of the Thompson groups.