| | | **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) | =m^{n}n!=
| C_{ m} wr S_{ n} | (cf. Corollary).
The general case is now easy:

**Corollary: **
*
If ** sÎS*_{ n} is of type *a=(a*_{1}, ...,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
* Å*^{n} 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.

last changed: January 19, 2005

| | | **Centralizers of elements in finite symmetric groups** |