# 3-transposition group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A **3-transposition group** is defined as a group that is either trivial or has a generating set that is a *single* conjugacy class whose elements are 3-transpositions, i.e., involutions such that the product of any of them with any conjugate has order 1, 2, or 3.

3-transposition groups were studied by Fischer. In addition to various infinite families, he identified a small number of them, called Fischer groups, that did not fall into the infinite families. This gave rise to three new sporadic simple groups.