Elementary abelian group

Summary

An elementary abelian $p$-group, for a prime $p$, is an abelian group in which every element other than the identity has order $p$. Such a group is the same thing as a vector space over the field $\F_p$. So a finite one is isomorphic to $(\Z/p\Z)^n$ for some $n$. They appear everywhere in finite group theory. For example, minimal normal subgroups of finite solvable groups are elementary abelian, and so is the quotient of a finite p-group by its Frattini subgroup.