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Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$

This induces a linear automorphism of $\mathbb{Z}[G]^G$, the algebra of $G$-invariant functions on $G$ under convolution, and surprisingly, this induced map is also an algebra automorphism, as can be seen by passing to $\mathbb{C}$ and noting that this is the Galois action on characters, which permutes the set of primitive idempotents of this algebra.

My question is whether this surprising fact can be explained directly, without using character theory? I would be interested in both a high-concept explanation of this symmetry, or a generators and relations argument for why there exists, for any $g,h$ in $G$, an $a,b\in G$ with: $$(gh)^n=ag^n a^{-1}bh^n b^{-1}.$$

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    $\begingroup$ Its true, and follows from the fact that $g\mapsto g^n$ is an algebra automorphism of $\mathbb{Z}[G]^G$, by expanding the product of the classes of $g$ and $h$. I don't know of a non character theoretic proof of why this map is actually an algebra automorphism however. $\endgroup$
    – Chris H
    Commented Feb 1, 2022 at 22:18
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    $\begingroup$ Wow, that's a beautiful find. I am getting a slightly cohomological whiff from it (as, e.g., in the proof of the transfer's homomorphism property). $\endgroup$
    – darij grinberg
    Commented Feb 1, 2022 at 23:58
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    $\begingroup$ The Frobenius group of order $20$ is an example, trivial outer automorphism group, with nonrational characters, take $n=3$. $\endgroup$
    – Chris H
    Commented Feb 2, 2022 at 4:54
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    $\begingroup$ There are a number of similar facts which seem non-obvious from a group-theoretic point of view, but are easy to prove with characters: for example, the number of times $ x \in G$ is expressible as a commutator is unchanged if we replace $x$ by a different generator of $\langle x \rangle.$ $\endgroup$
    – Geoff Robinson
    Commented Feb 2, 2022 at 11:20
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    $\begingroup$ By a generators and relations argument I would think some symbolic argument that holds in any torsion group of exponent coprime to $n$, building $a,b$ from $g,h$ using these assumptions and the assumed $n$th root function. If one could give a group of this type where the result fails, it would rule out such an argument, which would also be very interesting. $\endgroup$
    – Chris H
    Commented Feb 3, 2022 at 11:33

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