Let $p, q, r$ be distinct primes and let $n=p+q+r$. Show that $S_n$ contains a subgroup of order $pqr$.

Question

Let $p, q, r$ be distinct primes and let $n=p+q+r$. Show that $S_n$ contains a subgroup of order $pqr$.

I'm not sure how to start this, thanks!

Answer 1

Hint:

1. $S_p$ contains a subgroup of order $p$.
2. $S_p\times S_q$ is a subgroup of $S_{p+q}$.

Note: As @pjs36 mentioned, there's no need for $p,q,r$ being prime. The extra you got from that hypothesis is the subgroup you find is cyclic.

Answer 2

Think about the cyclic subgroup generated by the product of a $p$-cycle, a $q$-cycle, and a $r$-cycle, each disjoint with one another. What would the order of such an element, and the subgroup generated by it, be equal to?