Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.
Proof: Let $\sigma \in S_n$. Then $\sigma = (a_1a_2…a_{m_1})(a_{m_1+1}a_{m_1+2}…a_{m_2})….(a_{m_{k-1}+1}a_{m_{k-1}+2}…a_{m_{k}})$ represents the cycle decomposition of $\sigma$.
Suppose $|\sigma| = n$ is the order of an element in $S_n$.
So $\sigma^n = 1$.
Can someone please help me? I don't know how if I am doing this fine. And I am stuck.
Thank you for any help.
Best Answer
I don't think you should name $\color{red}n$ the order of an element in $S_{\color{red}n}$.
If $\sigma = \gamma_1\gamma_2\dotsm\gamma_k$ is a decomposition of the permutation $\sigma$ as a product of disjoint cycles, these cycles commute with each other. Hence $\sigma^r=\gamma_1^r\gamma_2^r\dotsm\gamma_k^r$. and $$\sigma^r=\gamma_1^r\gamma_2^r\dotsm\gamma_k^r=e\iff \gamma_1^r=\gamma_2^r=\dotsm=\gamma_k^r=e$$ whence $r$ is a common multiple of $o(\gamma_1), o(\gamma_2),\dots,o(\gamma_k)$. The smallest of these $r$s is by definition the least common multiple of the orders of the cycles.
On the other hand, the order of a cycle is no other than its length.