# Math Help - Disjoint cycles

1. ## Disjoint cycles

Prove that two disjoint cycles commute
Thanks

2. Let $\alpha, \beta \in S_n$ be two disjoint cycles. Clearly the $j \in \{1,2,...,n\}$ which are fixed by both are fixed by the product, in which ever order it is taken. Now consider those $j \in \{1,2,...,n\}$ which are not fixed by both. Since $\beta$ and $\alpha$ are disjoint, for a fixed $j \in \{1,2,...,n\}$ which is not fixed by both, exactly one of $\beta$ or $\alpha$ moves $j$. So, without loss of generality, let $\alpha(j)=k, \beta(j)=j$ with $j\neq k$. Since $k$ is part of the cycle $\alpha$, it must not be part of the cycle $\beta$ because $\beta$ and $\alpha$ are disjoint. Then $\alpha\beta(j)=\alpha(j)=k$ and $\beta\alpha(j)=\beta(k)=k$, so both $\beta\alpha$ and $\alpha\beta$ are the same function on $\{1,2,...n\}$.