TITLE: Are there any conditions , such that 2 permutations in $S_4$ are commutative?
QUESTION [11 upvotes]: It may a bit general question, but I'd like to ask anyway, and I'll try to detail as much as I can.
Well, I have the symmetric group, $S_4$. Lets take permutations $\sigma, \tau \in S_4 $.
So, $\sigma$ would look like this:
$$
        \begin{pmatrix}
        1 &2 &3 &4 \\
        \sigma_{1} &\sigma_{2} &\sigma_{3} &\sigma_{4} \\
        \end{pmatrix}
$$
And $\tau$ would look like this:
$$
        \begin{pmatrix}
        1 &2 &3 &4 \\
        \tau_{1} &\tau_{2} &\tau_{3} &\tau_{4} \\
        \end{pmatrix}
$$
my question is:
Are there any terms for $\sigma$ and $\tau$, that make their action commutative?
I.e Are there any terms such that $\sigma \star \tau = \tau \star \sigma$ ?
I hope I could be understandable. thank-you.

REPLY [14 votes]: There is a condition which classifies when two permutations commute. However, it is easier to state if we write the permutations using disjoint cyclic notation (see wikipedia).
So, write $\sigma=(a_{1, 1} \: a_{2, 1}\: \ldots \: a_{n_1, 1})\cdots (a_{1, i} \: a_{2, i}\: \ldots \: a_{n_i, i})$, then $\sigma$ and $\tau$ commute if and only if $(\tau(a_{1, 1}) \: \tau(a_{2, 1})\: \ldots \: \tau(a_{n_1, 1}))\cdots (\tau(a_{1, i}) \: \tau(a_{2, i})\: \ldots \: \tau(a_{n_i, i}))=\sigma$ (so apply $\tau$ to each of the "numbers" in $\sigma$).
For example, if $\sigma$ is a single cycle $(a_{1} \: a_{2}\: \ldots \: a_{n})$ then $\sigma$ and $\tau$ commute if and only if $(\tau(a_{1}) \: \tau(a_{2})\: \ldots \: \tau(a_{n}))$ is a cyclic shift of $(a_{1} \: a_{2}\: \ldots \: a_{n})$.
This follows from the fact that, in general, $$\tau^{-1}(a_{1, 1} \: a_{2, 1}\: \ldots \: a_{n_1, 1})\cdots (a_{1, i} \: a_{2, i}\: \ldots \: a_{n_i, i})\tau$$$$=(\tau(a_{1, 1}) \: \tau(a_{2, 1})\: \ldots \: \tau(a_{n_1, 1}))\cdots (\tau(a_{1, i}) \: \tau(a_{2, i})\: \ldots \: \tau(a_{n_i, i}))$$ which is quite well-known but I will leave proving it to the reader (it is in most advanced undergraduate texts). I will also leave interpreting this criterion to non-disjoint-cycle notation to the OP or any other interester person. I am hungry and have to run home for my dinner, so I have no time to do this myself!