There is an answer here, but it is a "roadmap". group containing normal subgroups of orders $3$ and $5$ contains element of order $15$ There are answers here, but they are "roadmaps" too. If $G$ contains normal subgroups of prime orders $p$ and $q$, then $G$ contains an element of order $pq$.
In the proofs given by Brian Bi linked here and by Alec Mouri linked here, it is concluded that $HK$ is a subgroup and is isomorphic to $H \times K$ and then that $HK$ contains an element of order 15.
Is it correct that it is either:
that the reason $HK$ contains an element of order 15 is that $HK$ is cyclic with order 15 or
that that's not reason $HK$ contains an element of order 15, but $HK$ is still cyclic with order 15?
Actually, are groups isomorphic to cyclic groups also cyclic with the same order?
I know if images of cyclic groups under surjective homomorphisms are cyclic, but I didn't think they would be of different orders if we also assume injective.
The following are screenshots of the proofs mentioned above (Kiefer Sutherland's voice)