I was wondering if my proof is adequate enough, or if there is a need for improvement. Any help appreciated!
Problem: Let $\alpha$ and $\beta$ be two disjoint cycles in $S_n$. Prove that $\alpha\beta$ = $\beta\alpha$.
My proof:
Let $\alpha$ = $(a_1, a_2, ..., a_r)$ and $\beta = (b_1, b_2, ..., b_s)$
Also let $x$ be any $a_r$ or $b_s$
So if $x$ = $a_r$, then $x\alpha\beta$ = $a_r\alpha\beta$ = $a_{r+1}\beta$ = $a_{r+1}$
And $x\beta\alpha$ = $a_r\beta\alpha$ = $a_r\alpha$ = $a_{r+1}$
Similiarily if $x$ = $b_s$, then $x\alpha\beta$ = $b_{s+1}$ = $x\beta\alpha$
So $x\alpha\beta$ = $x\beta\alpha$
Therefore $\alpha\beta$ = $\beta\alpha$