I have the following exercise with permutations:
Determine all the permutations $\sigma \in S_6$ which commute with $\alpha = (1 \ 2 \ 3)(4 \ 5 \ 6)$.
And the solution looks like this: $σα = ασ ⇐⇒ σασ^{-1} = α ⇐⇒ (σ(1), σ(2), σ(3))(σ(4), σ(5), σ(6)) = (1, 2, 3)(4, 5, 6) ⇐⇒ \{(σ(1), σ(2), σ(3)), \ (σ(4), σ(5), σ(6))\} = \{(1, 2, 3), \ (4, 5, 6)\}$.
But having this information, how do I determine the proper permutations? Sorry, I'm a begginer with these.
From the information given we have that $(\sigma(1),\sigma(2),\sigma(3)) = (1,2,3)$ or $(4,5,6)$. Now the cycles is completely determined by one element. So we have three options for $\sigma(1)$. If $\sigma(1) = 1$, then $\sigma(2) = 2$, $\sigma(3) = 3$. To find the complete permutation now you need to do the same for $(\sigma(4),\sigma(5),\sigma(6))$ and equate it to the other $3-$cycles. In a similar manner if you choose $\sigma(4) = 6$, then $\sigma(5) = 4, \sigma(6) = 5$. This choise gives us the permutation $\sigma = (4,6,5)$.
Similarly you can get all possible permutation.