# Elementary abelian 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

### Symbol-free definition

An **elementary abelian group** is a group that satisfies the following equivalent conditions:

- It is an abelian characteristically simple group
- It is a restricted direct product of isomorphic subgroups, each being cyclic of prime order
- It is the additive group of a vector space over a prime field

## Relation with other properties

### Stronger properties

- Cyclic group of prime order viz. simple Abelian group

### Weaker properties

## Facts

### Minimal normal subgroups

Any minimal normal subgroup in a solvable group must be elementary Abelian. This follows by combining the fact that it must be Abelian with the fact that in any group, a minimal normal subgroup is always characteristically simple.