Elementary abelian group

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 direct product of isomorphic subgroups, each being cyclic of prime order
 * It is the additive group of a vector space over a prime field

Stronger properties

 * Cyclic group of prime order viz. simple Abelian group

Weaker properties

 * Homocyclic group
 * Abelian group
 * Characteristically simple group

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.