Here is the problem
Let $p$ be a prime. Show that an element has order $p$ in $S_n$ if and only if its cycle decomposition is a product of commuting $p$-cycles. Show by an explicit example that this need not be the case of p is not prime.
I was able to show the "only if" part easily enough. But I am having difficulty with the "if" direction. I suppose that $\sigma \in S_n$ has order $p$ yet $\sigma$ is not just the product of commuting $p$-cycles, which would mean that an $n$-cycle, with $n \neq p$, would appear in the product. But this is where I was unsure of how to proceed, so I consulted this. However, the author claims that "the cycle decomposition of $\sigma$ contains an $n$-cycle $\tau$ for some $n \neq p$, and we can write $\sigma = \tau \omega$." Why does the $n$-cycle $\tau$ appear at the right-end of $\sigma$? Why couldn't it be at the left-end, or even in the interior?
Best Answer
The cycles in a cycle decomposition of $\sigma$ (not just any product of cycles) are disjoint, hence they commute. So it doesn't matter if it's in the interior; you can always move it to wherever you need it to be.
For example: (123)(45)(678) = (123)(678)(45).