Conjugacy classes of a symmetric group are determined by cycle structure

Conjugate of an -cycle is an -cycle

Let where an -cycle with the form

where then the conjugate is given by

and is hence also a -cycle sym

Proof

Let . Then . For any , , so . Hence maps numbers of the form to , and leaves all others invariant. Thus

as claimed.

This is a lemma for Conjugacy classes of a symmetric group are determined by cycle structure.

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