A finitely generated group has only finitely many subgroups of a given index
Let be a finitely generated group and let be a positive
integer. Let be a subgroup of of index and consider the
action of on the coset space by right multiplication.
Label the cosets , with the coset labelled by .
This gives a homomorphism
. Now, if and
only if , that is, fixes the coset . Therefore, , and this is completely
determined by . Now let be a finite generating set
for .
Then is determined by the images of the generators
. There are choices for the image of each , so there are at most homomorphisms . Hence,
there are only finitely many possibilities for .
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 |
---|---|
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 |