[[Group theory MOC]]
# Elementary abelian group

An **elementary abelian group** is an [[abelian group]] in which the [[group order|order]] of every non-identity element is the same. #m/def/group 
Since this must must be a prime number $p$, it follows that every elementary abelian group is a [[p-group]],
and such groups may be considered a [[vector space]] over [[Modular arithmetic|$\mathbb Z_p$]].

> [!tip]+ Notation
> For a prime $p$ and $h \in \mathbb{N}$, the (unique) elementary abelian group of order $p^h$ is denoted simply by $p^h$, i.e.
> $$
> \begin{align*}
> p^h = (\mathbb{Z}_{p}^+)^h
> \end{align*}
> $$

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#state/tidy | #lang/en | #SemBr