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I'm asked to write out the cycle decomposition of each element of order 2 in $S_4$. However, from my understanding, a cycle decomposition is the product of disjoint cycles. But the elements in $S_4$ of order 2 are already products of disjoint cycles. For example,

How do I simplify (1 2) or (1 2)(3 4)?

Isn't the only way to write (1 2) as a product (1 2)=(1 2)(1 2)? Which is not disjoint and hence not a cycle decomposition? And isn't (1 2)(3 4) already a cycle decomposition?

Nate
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  • (12)(34) is just one way of representing that permutation. Fundamentally, the permutation is the function that sends 1 -> 2, 2->1, 3-> 4, 4->3. The question is really asking for the obvious thing. – hoyland Oct 05 '16 at 14:40
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    https://math.stackexchange.com/questions/1598644/cycle-decompositions-of-s-4/1598658, https://math.stackexchange.com/questions/311680/finding-the-number-of-elements-of-order-two-in-the-symmetric-group-s-4/311684 – Eman Yalpsid Oct 05 '16 at 14:46

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It sounds like the question is looking for you to identify the ones which are of order 2. It's more of an identification problem and not a decomposition problem.

Dennis Muhonen
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