Symmetric Groups and Commutativity

Question

I just finished my homework which involved, among many things, the following question:

Let $S_{3}$ be the symmetric group $\{1,2,3\}$. Determine the number of elements that commute with (23).

Now, solving this was unproblematic - for those interested; the answer is 2. However, it got me thinking whether or not there is a general solution to this type of question.Thus, my question is:

Let $S_{n}$ be the symmetric group $\{1,\dots,n\}$. Determine the number of elements in $S_{n}$ that commute with $(ij)$ where $1\leq i,j \leq n $.

Answer 1

You can use orbit stabilizer theorem. The conjugacy class of $(i j)$ has size $ n \choose 2$. Therefore, $n!/{n \choose 2}$ elements of $S_n$ commute with $(i j)$.

Answer 2

• Let $\pi, \phi\in S_{\Omega}$. Then $\pi, \phi$ are disjoint if $\pi$ moves $\omega\in \Omega$ then $\phi$ doesn't move $\omega$.

For example, $(2,3)$ and $(4,5)$ in $S_6$ are disjoint. indeed, $\{2,3\}\cap\{4,5\}=\emptyset$.

Theorem: If $\pi, \phi\in S_{\Omega}$ are disjoint then $\pi\phi=\phi\pi$.