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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]] and let $F$ be a field whose [[characteristic|0F6ES3BRYP]] does not divide the order $\\abs{G}$.\nLet $V$ be a [[representation|WYGRXCG2RN]] of $G$ over $F$, that is, a module over the [[group algebra|MNP2HBWB5Q]] $F[G]$.\nThen every $G$-invariant subspace $W$ of $V$ has a $G$-invariant complement $W'$ with $V = W \\oplus W'$.\nEquivalently, the group algebra $F[G]$ is a [[semisimple ring|NY25HXA478]].\n\n## Remarks\n\nTake any linear projection $p \\colon V \\to W$ onto $W$ and average it over the group:\n$$\\tilde{p}(v) = \\frac{1}{\\abs{G}} \\sum_{g \\in G} g \\, p(g^{-1} v).$$\nThe division by $\\abs{G}$ is allowed by the hypothesis on the characteristic, $\\tilde{p}$ is $G$-equivariant and is still a projection onto $W$, and $W' = \\ker \\tilde{p}$ is the required complement.\nFor example, every finite-dimensional complex representation of a finite group is a direct sum of irreducible representations.\nThe hypothesis is necessary: let $G = \\set{1, g}$ have order $2$ and $F = \\F_2$, acting on $F^2$ by swapping coordinates.\nThe line spanned by $(1, 1)$ is invariant, but it is the only invariant line, so it has no invariant complement.\n\n## Explanations\n\n### L1 human\n\nLet a finite group act linearly on a space, and suppose one can divide by the size of the group.\nThen an invariant piece can always be split off cleanly from the rest.\nSo each representation breaks into irreducible building blocks.\nThis is the reason why representations of finite groups over the complex numbers are so well behaved.\n\n### L2 human\n\nBy the Artin–Wedderburn theorem, Maschke's theorem gives $\\C[G] \\cong M_{n_1}(\\C) \\times \\cdots \\times M_{n_k}(\\C)$, which leads to $\\abs{G} = \\sum n_i^2$ and to character theory.\nThe converse also holds: if the characteristic of $F$ divides $\\abs{G}$, then $F[G]$ is not semisimple.\nIndeed, $N = \\sum_{g \\in G} g$ spans a nonzero two-sided ideal with $N^2 = \\abs{G} N = 0$.\nA semisimple ring has no such nilpotent ideal.\nThat case is the subject of modular representation theory.\nThe averaging argument also works for compact groups with the Haar integral in place of the sum.\n",
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      "wikidata": "Q656198",
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        "url": "https://en.wikipedia.org/wiki/Maschke%27s_theorem"
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  "link": "https://tamnd.github.io/math2/node/4SBZ4EMPBZ/",
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  "markdown": "https://tamnd.github.io/math2/node/4SBZ4EMPBZ.md",
  "msc": [
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    "16S34",
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  "name": "Maschke's theorem",
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  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4SB/4SBZ4EMPBZ.md",
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  "tag": "4SBZ4EMPBZ",
  "uri": "m2://4SBZ4EMPBZ",
  "url": "https://tamnd.github.io/math2/wiki/maschkes-theorem/",
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