Conjugation of a product of disjoint cycles

Question

For a certain question I have to do, I have to use the fact that if $a$ is any permutation in $S_n$ and $\gamma = (i_1 \cdots i_j)$ is a cycle, then $$a\gamma a^{-1} = (a(i_1) \cdots a(i_j)),$$ for some $K \leqslant S_4$ (I have to determine $aKa^{-1} $ for all $a \in S_4)$ where $K$ has some elements which are the product of 2-cycles. How would I apply the above to a product of 2-cycles (namely the right-hand side)?

Answer 1

Just use the fact that $\gamma \mapsto \gamma a \gamma^{-1}$ is a group inner automorphism. Which means that for a product of cycles $c_1, \dots, c_m$ you have

$$\gamma (c_1 \dots c_m) \gamma^{-1} = (\gamma c_1 \gamma^{-1}) \dots (\gamma c_m \gamma^{-1}).$$

Applied to $2$-cycles:

$$\gamma ((a_1 \ a_2) \dots \dots (r_1 \ r_2)) \gamma^{-1} = (\gamma(a_1) \ \gamma(a_2)) \dots (\gamma(r_1) \ \gamma(r_2)).$$