The order of an element in Sym(n) is the least common multiple of the signature of its cycle decomposition

Prove that the order of an element in S_n is the least common multiple of the lengths of the cycles in its cycle decomposition.


Let \sigma \in S_n have the form \sigma = \prod_I \sigma_i where the \sigma_i are pairwise disjoint and \sigma_i is an n_i-cycle. Let |\sigma| = m, and let \ell = \mathsf{lcm}_I\{n_i\}. Note that since the \sigma_i are disjoint, we have \sigma_i^m = 1; hence n_i|m for each i \in I. By the definition of least common multiple, then, \ell|m. Now we also have that \sigma^\ell = \prod_I \sigma_i^\ell = 1, so that m|\ell. Since m and \ell are both positive, m = \ell. \blacksquare

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