# Group generated by involutions

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

## Contents

## Definition

### Symbol-free definition

A group is said to be **generated by involutions** if it has a generating set all whose elements are involutions (elements of order two).

Typical examples are triangle groups, and various reflection groups.

## Relation with other properties

### Stronger properties

- Triangle group
- Real special orthogonal group: This is the content of the Cartan-Dieudonne theorem
- Finitary symmetric group, in particular the symmetric group on a finite set.
`For full proof, refer: Transpositions generate the finitary symmetric group` - Finite simple non-Abelian group

### Weaker properties

## Metaproperties

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property

View a complete list of quotient-closed group properties

### Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property

View other finite direct product-closed group properties