Let h $=($a$1$ a$2...$ am$)$ be an m$-$cycle in Sn$.$ Prove that, for any permutation g in Sn$,$

g$1$hg $=($ag

$1$ ag

$2...$ ag

m$).$

For instance,

$(123)1(12)(123)=(1(123)2(123))=(23).$

$($This proves that conjugate permutations have the same cycle structure, i$.$e$.,$ when written

as products of disjoint cycles, there are the same number of cycles of each length