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## Homework Statement

Show that an element has order 2 in ##S_n## if and only if its cycle decomposition is a product of commuting 2-cycles.

## Homework Equations

## The Attempt at a Solution

Suppose that ##\sigma \in S_n## and ##| \sigma | =2##. Let the cycle decomposition of ##\sigma## be the following: ##\sigma = c_1 c_2 \dots c_m##. Then ##| \sigma | = lcm (|c_1|, |c_2|, \dots , |c_m|) = 2##. This is the case only if the ##|c_i| = 1## or ##|c_i| = 2## with at least one such that ##|c_i| = 2##. Hence ##\sigma## is a product of disjoint 2-cycles.

\Now, suppose that the cycle decomposition of ##\sigma## is a product of commuting 2-cycles: ##\sigma = c_1 c_2 \dots c_m##, where ##|c_1| = \cdots = |c_m| = 2##. Then ##| \sigma | = lcm (|c_1|, |c_2|, \dots, |c_m|) = lcm (2,2, \dots, 2) = 2##.

Is this at all a correct proof? Is there a way to do this without assuming a permutation can be decomposed uniquely into disjoint cycles, or that the order of a permutation is the least common multiple of the orders of the cycles in is decomposition?