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## Homework Statement

Let ##n## be a natural number and let ##\sigma## be an element of the symmetric group ##S_n##. Show that if ##\sigma## is a product of disjoint cycles of orders ##m_1 , \dots , m_k##, then ##|\sigma|## is the least common multiple of ##m_1 , \dots , m_k##.

## Homework Equations

## The Attempt at a Solution

So if I suppose that ##\sigma = c_1 c_2 \dots c_k## where ##|c_i| = m_i##, and ##|\sigma| = n##, then ##\sigma ^n = c_1^n \dots c_k^n = 1##. Now, if I could show that this implies that ##c_i^n = 1## then I would basically be done, but I am not sure how to do this. It's kind of "obvious" to me since after taking powers we still have disjoint cycles, and also since they're disjoint none of the disjoint cycles can be inverses of other possible compositions of other disjoint cycles, so they must all be the identity.