Elementary abelian group C_p^1

Let $p$ be prime. The elementary abelian group $C_p^{1}$ is the additive group of $\mathbb F_p^{1}$ , with coordinatewise addition. Equivalently it is the direct product of $1$ copies of the cyclic group of order $p$ .

Order and simplicity

Its order is $p$ , and every nonzero element has order $p$ . It is cyclic of prime order and 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.