Symmetric Group - Conjugacy Classes

Conjugacy Classes

The conjugacy classes of S_n correspond to the cycle structures of permutations; that is, two elements of S_n are conjugate in S_n if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S₅, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S_n can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example:

which can be written as the product of cycles, namely: (2 4).

This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, i.e.

It is clear that such a permutation is not unique.