The exercise goes like this:
Find all the permutations which commute with $\sigma$ = (1 3 2 5 4) (6 7) in $S_7$.
I am using a standard approach for this kind of problem: $\tau \in S_7$ commutes with $\sigma$ if and only if $\tau\sigma\tau^{-1} =\sigma$. Thus we get:
$$ \tau\sigma\tau^{-1} = (\tau(1)\:\tau(3)\:\tau(2)\:\tau(5)\:\tau(4))\:(\tau(6)\:\tau(7)) = (1\:3\:2\:5\:4)\:(6\:7).$$
What I'm struggling with is how to get $\sigma$ from this equation. How does one determine it by its first element?
I'm a complete begginer with this topic and unfortunately there is not much info on permutations in my native language.
There is only one way to satisfy $$(\tau(1)\:\tau(3)\:\tau(2)\:\tau(5)\:\tau(4))\:(\tau(6)\:\tau(7)) = (1\:3\:2\:5\:4)\:(6\:7)$$ and that is if first $(\tau(1)\:\tau(3)\:\tau(2)\:\tau(5)\:\tau(4)) = (1\:3\:2\:5\:4)$ and second $(\tau(6)\:\tau(7)) = (6\:7)$.
We have $(\tau(6)\:\tau(7)) = (6\:7)$ if $\tau$ either fixes $6$ and $7$ or swaps them.
As for $(\tau(1)\:\tau(3)\:\tau(2)\:\tau(5)\:\tau(4)) = (1\:3\:2\:5\:4)$, once you pick $\tau(1) \in \{1,2,3,4,5\}$, this condition tells you what $\tau(2), \tau(3), \tau(4), \tau(5)$ must be.
For example, if $\tau(1) = 5$ then $\tau(3)$ must be $4$ (the next term in this $5$-cycle after $5$), $\tau(2)$ must be $1$ (the next term after $4$) and so on.