Centralizers of elements in finite symmetric groups

As an application of this permutation representation we obtain a description of the centralizers of elements in finite symmetric groups. To show this we note that , where , is just the centralizer of

This follows from , which is clear from together with and the fact that (cf. ). The general case is now easy:

. Corollary   If is of type , then is a subgroup of which is similar to the direct sum

Similarly we can show (recall )

. Corollary   The normalizer of the -fold direct sum

is conjugate to the plethysm

Thus centralizers of elements and normalizers of specific subgroups of symmetric groups turn out to be direct sums of complete monomial groups. Since such groups will also occur as acting groups later on, we also describe their conjugacy classes.