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Does the trivial representation always induce the permutation representation? Is this true for each field $\mathbb{K}$ or just for representations over $\mathbb{C}$?

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    The induction of the trivial representation from _a_ subgroup is _a_ permutation representation. A nontrivial group has many permutation representations, so you cannot speak of "_the_ permutation representation". But yeah, this is true over any field, pretty much by definition of induction.2012-04-19
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    Thanks!Is it also true, that the dimension of a permutation repr. comes from the index? (I mean the index of $G$ and the subgroup you use for induction)2012-04-19

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