Exercise 4.3.9 (Centralizer of $(1\ 2)(3\ 4)$ in $S_n$)

Proof. Let ${𝒪}_{\sigma }$ denote the orbit of $\sigma \in S_n$ for the action of conjugation, i.e. the conjugation class of $\sigma$.

We compute the cardinality of ${𝒪}_{(12)(34)}$.

By Proposition 11, the elements of ${𝒪}_{(12)(34)}$ are the permutations $(ab)(cd)$, where $a,b,c,d$ are distinct. We can choose $a$ among $n$ elements $1,2,\dots n$, then we choose $b$ among the $n-1$ elements distinct of $a$, $c$ among $n-2$ elements and $d$ among $n-3$ elements. This gives $n(n-1)(n-2)(n-3)$ choices.

Moreover

so we must divide this result by $8$ to obtain

By Proposition 6,

so, if $n\ge 4$,

Note that every permutation which fixes the elements $1,2,3,4$ commutes with $(12)(34)$. This gives $(n-4)!$ permutations of $S_{[[5,n]]}$ in $C_{S_n}((12)(34))$.

Moreover, we can check that the $8$ following permutations in the set $A$ commute with $(12)(34)$:

Since the support of these $8$ permutations is included in $\{1,2,3,4\}$, every permutation of the form $\mathit{𝜎𝜏}$, where $\sigma \in A$ and $\tau \in S_{[[5,n]]}$ is in the centralizer $C_{S_n}((12)(34))$, and there are $8\cdot (n-4)!$ such permutations. By the order computation above, this is the full centralizer of $(12)(34)$ in $S_n$:

where

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