Conjugacy Classes
The conjugacy classes of Sn correspond to the cycle structures of permutations; that is, two elements of Sn are conjugate in Sn 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 Sn 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.
Read more about this topic: Symmetric Group
Famous quotes containing the word classes:
“Is a man too strong and fierce for society, and by temper and position a bad citizen,—a morose ruffian, with a dash of the pirate in him;Mnature sends him a troop of pretty sons and daughters, who are getting along in the dame’s classes at the village school, and love and fear for them smooths his grim scowl to courtesy. Thus she contrives to intenerate the granite and the feldspar, takes the boar out and puts the lamb in, and keeps her balance true.”
—Ralph Waldo Emerson (1803–1882)