Find $ [ S_9 : H ]$

Question

Consider the permutation $\sigma = (1,2)(4,7,8)(2,1)(7,2,8,1,5)(5,6) \in S_9$ . Let $H= \langle \sigma \rangle $. Find $ [ S_9 : H ]$

I know to find $ [ S_9 : H ]$ you take $|G|/|H|$ and $|G|= 9!= 362880$, but what is $|H|$? I know it's $\langle \sigma \rangle$, but it's unclear to me what the number is suppose to be, is it just 5? Making $ [ S_9 : H ] = 72576$?

Answer 1

Hint: You can write $\sigma$ using disjoint cycle notation, meaning you can write it as a product of cycles that do not share any elements. Then these cycles don't affect each other and you understand the permutation entirely from that form. If $\sigma=C_1C_2\cdots C_n$ where the $C_i$'s are disjoint cycles, then $|\sigma|=|H|=\mathrm{lcm}(|C_1|,|C_2|\cdots |C_n|)$. As a further hint, note that between $(1,2)$ and $(2,1)$, there are no cycles that use $2$ or $1$, so the two effectively cancel each other out, so $\sigma=(4,7,8)(7,2,8,1,5)(5,6)$.

Answer 2

Hint: The order of $H$ is the order of $\sigma$, which is the lcm of the order of the cycles in its decomposition into disjoint cycles.