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arXiv · 1105.6066

On the divisibility of $#\Hom(\Gamma,G)$ by $|G|

Fernando Rodriguez Villegas·Cameron Gordon
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Abstract

We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $\Gamma$ is a finitely generated group, then $|G|$ divides $#\Hom(\Gamma,G)$ for every finite group $G$ if and only if $\Gamma$ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group $\Gamma$.

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Fernando Rodriguez Villegas, Cameron Gordon. 2011-05-30. On the divisibility of $#\Hom(\Gamma,G)$ by $|G|. https://arxiv.org/abs/1105.6066

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