Consider the dihedral group $D_{182}$ of order $182=2\cdot 91$.
I am to show that this group doesn't contain a cyclic subgroup of order $14=2\cdot 7$.
My approach: The textbook states that for a group G of order $pq$ (for primes $p$ and $q$, with $p<q$), if $p$ divides $q-1$, there is a unique non-abelian group of order $pq$.
Since 2 and 7 are primes, and $2<7$, I can show that since 2 divides $7-1$, the subgroup of order 14 must be unique and non-abelian. Since all cyclic groups are abelian, this subgroup can't be cyclic. And since it is unique, no other subgroups of order 14 exist.
Is this correct and sufficient? The exercise hints that information from a completely different chapter can be used, so I wonder if I've missed something.
Thanks in advance!