Let $G$ be a finitely generated group and let $n$ be a positive integer. Let $H$ be a subgroup of $G$ of index $n$ and consider the action 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 and only if $Hx = H$ , that is, $G$ fixes the coset $H$ . Therefore, $H = \stab_G(1) = \{g \in G \mid 1(g\phi) = 1\}$ , and this is completely determined 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 $H$ .

Bibliography

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