Let $X$ be any finite set and $G$ be a subgroup of permutations of $X$. Let $F$ be the complex vector space of complex valued functions defined in $X$. Consider the action $(g.f) (x) =f(g^{-1}(x))$ Then there is an element $\sigma$ of $F$ such that $g.\sigma=\sigma$ for all $g$ in $G$. Also $F$ can be written as $H+c(\sigma)$ where $H$ is $G$-invariant subspace. I got this question from one of my friends. Now if $G$ is the whole permutation group of $X$ then I am able to show that there exists a $\sigma$ of $F$ such that $g.\sigma$ . But what is the proof when $G$ is a subgroup of permutation group of $X$? Thank you in advance.
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$\begingroup$ there exists $\sigma$ of $F$ such that what? In the 4th sentence from the end. $\endgroup$– WolframJan 29, 2017 at 18:24
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$\begingroup$ All constant functions are invariant. $\endgroup$– anonJan 30, 2017 at 3:34
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