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

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

2011-05-30

arxiv.org/abs/1105.6066v1

Mathematics

Group Theory

Scientific paper

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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