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 .
|Title||A finitely generated group has only finitely many subgroups of a given index|
|Date of creation||2013-03-22 15:16:03|
|Last modified on||2013-03-22 15:16:03|
|Last modified by||avf (9497)|