Conjugacy classes of a symmetric group are determined by cycle structure

Two permutations are conjugate iff they have the same number of -cycles for each . #m/thm/group

Proof

Let where the product of disjoint cycles

Then conjugating by is the same as the product of conjugating each cycle

but The conjugate of an -cycle is an -cycle, hence the cycle structure of is identical.

The conjugacy classes of thus correspond to partitions of .

#state/tidy | #lang/en | #SemBr