How to show the length of a concatenated cycle?

Question

I have been asked:

And I am stuck. I have thought about concatenating the permutation and analyzing the length of the result, but that is not leading anywhere.

Ideally I don't want the answer, just a hint as to how you can begin to show what the length of the resulting permutation is.

Answer 1

Hint: Let $\vartheta:=\sigma\gamma\sigma^{-1}$ and express $\vartheta(\, \sigma(a_i)\,)$ for $i=1..n$.
Then check what's $\vartheta(b)$ for $b\ne \sigma(a_i)$.