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Suppose $G$ is a finite group and that $E$ is an equation of the form $x_1 x_2 ... x_n = e$, where each $x_i$ is in the set of symbols $\{x, y, x^{-1}, y^{-1}\}$.

Is it always true that the number of ordered pairs $(g, h)$ of elements of $G$ satisfying $E$ is a multiple of $|G|$?

I know that it is true in many cases and I can't find any counterexamples. Does anybody know a simple counterexample, or a suitable reference for this question?

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    $\begingroup$ @M.Vinay Thanks for that - it looks much better! $\endgroup$
    – Paul Boddington
    Commented Feb 24, 2015 at 1:11

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Yes. This is a special case of results in http://arxiv.org/pdf/1205.2824.pdf

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    $\begingroup$ Indeed, it appears to be a special case of results in Louis Solomon, The solution of equations in groups, Arch. Math. 20 (1969) 241-247, MR0249497 (40 #2742). $\endgroup$
    – Gerry Myerson
    Commented Feb 24, 2015 at 4:52

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