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  "body": "\n## Definition\n\nLet $p$ be a [[prime|01826KMG2H]].\nA group $A$ is an *elementary abelian $p$-group* if $A$ is [[abelian|082GHNPJ53]] and every element $a \\neq e$ of $A$ has [[order|0WE2NHZJ7B]] $p$.\nA group is *elementary abelian* if it is an elementary abelian $p$-group for some prime $p$.\n\n## Remarks\n\nExample: $(\\Z/3\\Z)^2$ is elementary abelian: it is abelian, and $3(a, b) = (3a, 3b) = (0, 0)$, so every nonzero element has order $3$.\n\nNon-example: $\\Z/4\\Z$ is abelian but not elementary abelian, since $1$ has order $4$.\nNon-example: the symmetric group $S_3$ is not elementary abelian: it is not abelian, and it has elements of orders $2$ and $3$.\n\nWriting $A$ 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.\nHence a finite elementary abelian $p$-group is isomorphic to $(\\Z/p\\Z)^n$, where $n$ is its dimension, and its order is $p^n$.\nIts automorphism group is the [[general linear group|0FCHE902D7]] $\\mathrm{GL}_n(\\F_p)$.\nThe trivial group is elementary abelian for every $p$, with $n = 0$.\n",
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