A finitely generated group has only finitely many subgroups of a given index

Let $G$ be a finitely generated group and let $n$ be a positiveinteger. Let $H$ be a subgroup of $G$ of index $n$ and consider theaction of $G$ on the coset space $(G:H)$ by right multiplication.Label the cosets $1,\\ldots,n$ , with the coset $H$ labelled by $1$ .This gives a homomorphism $\\phi:G\\to S_{n}$ . Now, $x\\in H$ if andonly if $Hx=H$ , that is, $G$ fixes the coset $H$ . Therefore, $H=\\mathrm{Stab}_{G}(1)=\\{g\\in G\\mid 1(g\\phi)=1\\}$ , and this is completelydetermined by $\\phi$ . Now let $X$ be a finite generating set for $G$ .Then $\\phi$ is determined by the images $x\\phi$ of the generators $x\\in X$ . There are $|S_{n}|=n!$ choices for the image of each $x\\in X$ , so there are at most $n!^{|X|}$ homomorphisms $G\\to S_{n}$ . Hence,there are only finitely many possibilities for $H$ .

References

1 M. Hall, Jr., A topology for free groups and related groups, Ann. ofMath. 52 (1950), no. 1, 127–139.

单词	AFinitelyGeneratedGroupHasOnlyFinitelyManySubgroupsOfAGivenIndex
释义	A finitely generated group has only finitely many subgroups of a given index Let $G$ be a finitely generated group and let $n$ be a positiveinteger. Let $H$ be a subgroup of $G$ of index $n$ and consider theaction of $G$ on the coset space $(G:H)$ by right multiplication.Label the cosets $1,\\ldots,n$ , with the coset $H$ labelled by $1$ .This gives a homomorphism $\\phi:G\\to S_{n}$ . Now, $x\\in H$ if andonly if $Hx=H$ , that is, $G$ fixes the coset $H$ . Therefore, $H=\\mathrm{Stab}_{G}(1)=\\{g\\in G\\mid 1(g\\phi)=1\\}$ , and this is completelydetermined by $\\phi$ . Now let $X$ be a finite generating set for $G$ .Then $\\phi$ is determined by the images $x\\phi$ of the generators $x\\in X$ . There are $\|S_{n}\|=n!$ choices for the image of each $x\\in X$ , so there are at most $n!^{\|X\|}$ homomorphisms $G\\to S_{n}$ . Hence,there are only finitely many possibilities for $H$ . References 1 M. Hall, Jr., A topology for free groups and related groups, Ann. ofMath. 52 (1950), no. 1, 127–139.
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