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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?