How many subgroups of $\Bbb Z_5\times \Bbb Z_6$ are isomorphic to $\Bbb Z_5\times \Bbb Z_6$ I am trying to find the answer to the question in the title. The textbook's answer is only $\Bbb Z_5\times \Bbb Z_6$ itself. But i think like the following: Since 5 and 6 are relatively prime, $\Bbb Z_5\times \Bbb Z_6$ is isomorphic to $\mathbb{Z}_{30}$. And also, since 2,3 and 5 are pairwise relatively prime, then $\Bbb Z_2\times \Bbb Z_3\times \Bbb Z_5$ should also be isomorphic to $\Bbb Z_5\times \Bbb Z_6$. Am i wrong?
Thank you

The underlying sets of these groups are by definition $$\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5=\{(a,b,c):a \in \mathbb{Z}_2 \text{ and } b \in \mathbb{Z}_3 \text{ and } c \in \mathbb{Z}_5\}$$ and $$\mathbb{Z}_5 \times \mathbb{Z}_6=\{(a,b):a \in \mathbb{Z}_5 \text{ and } b \in \mathbb{Z}_6\}.$$  So while $\mathbb{Z}_5 \times \mathbb{Z}_6$ is trivially a subgroup of $\mathbb{Z}_5 \times \mathbb{Z}_6$, the group $\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5$ is not even a subset of $\mathbb{Z}_5 \times \mathbb{Z}_6$.  In fact, they have no elements in common.
However, the groups $(\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_5,+)$ and $(\mathbb{Z}_5 \times \mathbb{Z}_6,+)$ are isomorphic (they have essentially the same structure).  In fact, they are isomorphic to infinitely many other groups.
