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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 $\mathbb{Z}_{75}$?

Hydrogen
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    What are the orders of the elements of $\Bbb Z_5 \oplus \Bbb Z_{15}$? In particular, does that group have any element of order $75$? – Travis Willse May 16 '22 at 15:35
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    It hasn't got an element of order $75$. Your broader question, however, is not generally solvable. Even the simpler sounding question, "given the generators and relations that define a group, can I tell if the group is trivial?" is too hard. – lulu May 16 '22 at 15:35
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    @TravisWillse, Thank you. – Hydrogen May 16 '22 at 16:31
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    Cf. https://math.stackexchange.com/questions/955577/mathbbz-mathbb2z-bigoplus-mathbbz-mathbb2z-not-isomorphic-to-ma – Travis Willse May 16 '22 at 16:39

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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 $\mathbb{Z}_5 \oplus \mathbb{Z}_{15}$

mikecat1024
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  • I'm not quite sure why the order of $y$ is $3$ or $5$. The order is the lowest $k\in\mathbb{N}$ such that $k\cdot y=0$. But if we consider $y=13$, then $3\cdot y=9$ and $5\cdot y=5$. The order of $13$ is $15$, if I'm not wrong? – Hydrogen May 16 '22 at 16:19
  • I do agree, however, that the highest order of $(x,y)$ is 15. – Hydrogen May 16 '22 at 16:22
  • Sorry, I missed. The order of $(x,y)$ is $1$, $3$, $5$ or $15$. – mikecat1024 May 16 '22 at 16:37