Elementary abelian group C_p^n

Let $p$ be prime and let $n\geq1$ be an integer. The elementary abelian group $C_p^{n}$ is the additive group of $\mathbb F_p^{n}$ , with coordinatewise addition. Equivalently it is the direct product of $n$ copies of the cyclic group of order $p$ .

Order and simplicity

Its order is $p^{n}$ , and every nonzero element has order $p$ . The case $n=1$ is cyclic of prime order and simple. For $n>1$ , a one-dimensional subspace is proper nontrivial normal, so simplicity fails.

Structure being selected

This record selects the additive group structure. Linear maps over $\mathbb F_p$ and group homomorphisms between elementary abelian $p$ -groups agree, because repeated addition supplies scalar multiplication by every element of the prime field.