[Math] Expression as a product of disjoint cycles

Question

Let $\alpha = (9312)(496)(37215) \in S_n, n \ge 9$. Express $\alpha$ as a product of disjoint cycles.

I know this is probably a really easy question, but my professor didn't elaborate on how to exactly do this and neither does my assigned text. If anyone could elaborate on the algorithm of going about this I would really appreciate it.

Thanks you

Answer 1

Start from $1$, compute its image, compute the image of this image, and so on until this goes back to $1$. Iterate starting from the smallest element not met until then.

Here $1\to5\to1$ hence your first cycle is $(15)$. Next, $2\to2$. Next, $3\to7\to9\to6\to4\to3$ hence another cycle is $(37964)$. You are left with another fixed point $8\to8$.

Thus, $\alpha=(15)(37964)$.