How to show that two groups are not isomorphic.

Question

I have learned various theorems that tell me when two groups are isomorphic. For example, if the greatest common divisor of $j$ and $k$ is equal to one, then $\mathbb{Z}_j\oplus\mathbb{Z}_k\cong\mathbb{Z}_{jk}$. This tells us, for instance, that $\mathbb{Z}_3\oplus\mathbb{Z}_{25}\cong\mathbb{Z}_{75}$. However, how can I show that $\mathbb{Z}_5\oplus\mathbb{Z}_{15}$ is not isomorphic to $\mathbb{Z}_{75}$?

Answer 1

If $\mathbb{Z}_{75}$ is isomorphic to $\mathbb{Z}_5 \oplus \mathbb{Z}_{15}$, $\mathbb{Z}_5 \oplus \mathbb{Z}_{15}$ has a element of order $75$.

But for all $(x,y) \in \mathbb{Z}_5 \oplus \mathbb{Z}_{15}$, the order of $x$ is $5$ and the order of $y$ is $1$, $3$, $5$, or $15$. Therefore, the order of $(x,y)$ is at most $15$.

This is contradiction. Hence, $\mathbb{Z}_{75}$ is not isomorphic to $\mathbb{Z}_5 \oplus \mathbb{Z}_{15}$