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 $\{x, y, x^{-1}, y^{-1}\}$.



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



Number of solutions to equations in finite groups
Yes. This is a special case of results in <http://arxiv.org/pdf/1205.2824.pdf>


