# Elementary abelian 2-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

## Contents

## Definition

An **elementary abelian 2-group** or **Boolean group** is a group satisfying the following equivalent conditions:

- It is isomorphic to the additive group of a vector space over the field of two elements.
- It is either the trivial group or is isomorphic to the additive group of a field of characteristic two.
- It is a group of exponent at most two.
- All the non-identity elements of the group are involutions.

### Equivalence of definitions

`Further information: Exponent two implies abelian`