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  "body": "\n## Summary\n\nAn elementary abelian $p$-group, for a prime $p$, is an [[abelian group|082GHNPJ53]] in which every element other than the identity has order $p$.\nSuch a group is the same thing as a [[vector space|000AYPA9AJ]] over the field $\\F_p$.\nSo a finite one is isomorphic to $(\\Z/p\\Z)^n$ for some $n$.\nThey appear everywhere in finite group theory.\nFor example, minimal normal subgroups of finite solvable groups are elementary abelian, and so is the quotient of a finite [[p-group|06QYRTX9KQ]] by its Frattini subgroup.\n\n## Explanations\n\n### L1 human\n\nTake $n$ switches, each of which can be on or off, and combine two settings by flipping the switches that are on in the second one.\nDoing the same move twice brings you back to the start, so every element has order $2$.\nThis is the elementary abelian group $(\\Z/2\\Z)^n$ with $2^n$ elements.\nFor $n = 2$ it is the Klein four-group.\nReplacing on/off by the $p$ residues modulo a prime $p$ gives $(\\Z/p\\Z)^n$.\n",
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