Elementary abelian group

Remarks

Example: $(\Z/3\Z)^2$ is elementary abelian: it is abelian, and $3(a, b) = (3a, 3b) = (0, 0)$, so every nonzero element has order $3$.

Non-example: $\Z/4\Z$ is abelian but not elementary abelian, since $1$ has order $4$. Non-example: the symmetric group $S_3$ is not elementary abelian: it is not abelian, and it has elements of orders $2$ and $3$.

Writing $A$ additively, the scalar multiplication $(k + p\Z) \cdot a = ka$ is well defined because $pa = 0$, and it makes $A$ a vector space over the finite field $\F_p$; the subgroups are the subspaces. Hence a finite elementary abelian $p$-group is isomorphic to $(\Z/p\Z)^n$, where $n$ is its dimension, and its order is $p^n$. Its automorphism group is the general linear group $\mathrm{GL}_n(\F_p)$. The trivial group is elementary abelian for every $p$, with $n = 0$.