Is $H$ a subgroup?

Question

For fixed $a,b$ in $S_n$ let $H=\{a^ib^j \mid i,j \text{ integers} \}$. Is $H\le S_n$?

Answer 1

This is not true. Consider $n=5$, $a=(1\,2\,3)$, $b=(3\,4\,5)$. Then $H$ has at most $9$ elements and contains the element $ab=(1\,2\,3\,4\,5)$ of order $5$ as well as $a$ of order $3$, hence must have at least $15$ elements to be a group.

Answer 2

The smallest counterexample is $n=3$, $a=(12)$, and $b=(13)$.