When talking about permutations such as $(1\ 2 \ 3\ 4\ 5)$ and $(5\ 4\ 3\ 2\ 1\ )$, do permutations and their inverses always have the same cycle type? If so, is there a proof of this and why does it work?
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2026-03-25 15:39:50.1774453190
Do permutations and their inverses always have the same cycle type?
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Yes, because when you write a permutation $p$ as a product $c_1c_2\ldots c_k$ of disjoint cycles, they commute (since they're disjoint). So, $p^{-1}={c_1}^{-1}{c_2}^{-1}\ldots {c_k}^{-1}$.