Rules associated with disjoint cycle operations

Question

I have two permutations (expressed using disjoint cycle notation), $f=(1 8 7 4 9 6 3 2 5)$ and $g=(137)(26548)(9)$, and am trying to calculate $g^f=f^{-1}gf$.

My question is whether $((137)(26548)(9))^f=(137)^f(26548)^f(9)^f$, and, if it does, whether somebody could prove this (or at least give me some intuition behind it).

Thanks for any help,

Jack

Answer 1

The answer is affirmative, and to understand this let's try a general proof: let's show that $(gh)^f = g^f h^f$. We have

$$(gh)^f = f^{-1}(gh)f = f^{-1}(geh)f = f^{-1} \Big( g (f f^{-1}) h \Big) f = (f^{-1}gf) (f^{-1}hf) = g^f h^f .$$

In more formal terms we express this by saying that "conjugation (by $f$) is a group homomorphism".