Symmetric group

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In mathematics, the symmetric group on n symbols, Sn, is the group composed by all permutations of n symbols.

It has n! elements (as many as the permutations of n elements).

Its conjugacy classes correspond to the cycle structures of permutations; that is, two elements of Sn are conjugate if and only if they consist of the same number of cycles of the same lengths. For instance, in S5, (123)(45) and (143)(25) are conjugate; (123)(45) and (12)(3)(45) are not.