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 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 $1,\mathrm{\dots},n$, with the coset $H$ labelled by $1$. This gives a homomorphism^{} $\varphi :G\to {S}_{n}$. Now, $x\in H$ if and only if $Hx=H$, that is, $G$ fixes the coset $H$. Therefore, $H={\mathrm{Stab}}_{G}(1)=\{g\in G\mid 1(g\varphi )=1\}$, and this is completely determined by $\varphi $. Now let $X$ be a finite generating set^{} for $G$. Then $\varphi $ is determined by the images $x\varphi $ 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. of Math. 52 (1950), no. 1, 127–139.
Title | A finitely generated group has only finitely many subgroups of a given index |
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Canonical name | AFinitelyGeneratedGroupHasOnlyFinitelyManySubgroupsOfAGivenIndex |
Date of creation | 2013-03-22 15:16:03 |
Last modified on | 2013-03-22 15:16:03 |
Owner | avf (9497) |
Last modified by | avf (9497) |
Numerical id | 6 |
Author | avf (9497) |
Entry type | Theorem |
Classification | msc 20E07 |
Related topic | Group |
Related topic | FinitelyGenerated |