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I was wondering if anyone here could pitch a plausible geometric interpretation of Maschke's Theorem for $FG$-modules (or at least for a particular instance of its conclusions.) It seems reasonable that there should be some sort of picture associated with averaging a projection over a finite group, but the few examples that have occurred to me seem uninteresting; when is the direct sum decomposition in question sensible but not obvious?

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    That does seem to be a very natural way of thinking about it. The proof I first came across avoids any mention of the standard inner product, but I guess the process of passing from an ordinary complementary subspace (such as $V^{\perp}$) to the kernel of an $FG$-homomorphism is basically the same. Thanks!2011-10-17

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