---
tag: 8T34EC16NH
kind: definition
name: Elementary abelian group
msc: [20K01, 20D15]
defines: DZSZ8ZS554
uses: [01826KMG2H, 082GHNPJ53, 0WE2NHZJ7B, 000AYPA9AJ, 0ZXZ82FTX2, 0FCHE902D7]
sources:
  - url: https://en.wikipedia.org/wiki/Elementary_abelian_group
---

## Definition

Let $p$ be a [[prime|01826KMG2H]].
A group $A$ is an *elementary abelian $p$-group* if $A$ is [[abelian|082GHNPJ53]] and every element $a \neq e$ of $A$ has [[order|0WE2NHZJ7B]] $p$.
A group is *elementary abelian* if it is an elementary abelian $p$-group for some prime $p$.

## 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|000AYPA9AJ]] over the [[finite field|0ZXZ82FTX2]] $\F_p$; 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|0FCHE902D7]] $\mathrm{GL}_n(\F_p)$.
The trivial group is elementary abelian for every $p$, with $n = 0$.