Is correct to say $H\times K$ is a subgroup of $G$ and its order is $q$? [closed]
Let $G\cong \mathbb{Z}_n\times \mathbb{Z}_m$.
If I proved $H\times K$ is a subgroup of $\mathbb{Z}_n\times \mathbb{Z}_m$ and $|H\times K|=q$,
is correct to say $H\times K$ is a subgroup of $G$ and its order is $q$?
Solution 1:
No, not exactly, since the (underlying set of) each subgroup of $G$ is a subset of (the underlying set of) $G$.
The best you can say is that $H\times K$ is isomorphic to a subgroup of $G$.
Isomorphisms preserve order.
All this is true regardless of what $G,H,K$ are.