Abstract Algebra I · Cyclic and permutation groups
Disjoint cycles commute
Problem
Let $\sigma,\tau\in S_n$ have disjoint supports. Prove $\sigma\tau=\tau\sigma$. Explain why merely being written as separate cycles is not enough unless disjointness is checked.
Hint
Fix a symbol $x$ and split into membership in the support of $\sigma$, the support of $\tau$, or neither.
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