Centralizers of elements in finite symmetric groups

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 d[ C _m wr S _n ], where C _m := á(1 ...m) ñ, is just the centralizer of

s:=(1 ...m)(m+1, ...,2m) ...((n-1)m+1, ...,nm) ÎS _mn .

This follows from d[ C _m wr S _n ] ÍC_S( s), which is clear from the formula together with the formula and the fact that | C_S( s) | =mⁿn!= | C _m wr S _n | (cf. Corollary). The general case is now easy:

Corollary: If sÎS _n is of type a=(a₁, ...,a_n), then C_S( s) is a subgroup of S _n which is similar to the direct sum

Å_i(C _i S _ai).

Similarly we can show (recall the examples)

Corollary: The normalizer of the n-fold direct sum

Åⁿ S _m := S _m Å...ÅS _m , n summands,

is conjugate to the plethysm S _m S _n .

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.

Centralizers of elements in finite symmetric groups