Elementary abelian group C_p^3

Let $p$ be prime. The elementary abelian group $C_p^{3}$ is the additive group of $\mathbb F_p^{3}$ , with coordinatewise addition. Equivalently it is the direct product of $3$ copies of the cyclic group of order $p$ .

Order and simplicity

Its order is $p^{3}$ , and every nonzero element has order $p$ . Any one-coordinate subspace is a proper nontrivial subgroup. Because the group is abelian, that subgroup is normal, so the group is not simple.

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.