2
$\begingroup$

Prove that $\mathbb{Z}_4 \times \mathbb{Z}_{18}\times \mathbb{Z}_{15}$ and $\mathbb{Z}_3 \times \mathbb{Z}_{36}\times \mathbb{Z}_{10}$ are isomorphic groups.

Here order of the 2 groups are same. Also I know that $Z_m \times Z_n$ is isomorphic to the cyclic group $Z_{mn}$ iff $(m,n)=1$. But I don't know to find the group which is isomorphic to $Z_m \times Z_n$ where m and n are not relatively prime.

$\endgroup$
4
$\begingroup$

$$\begin{align*} \mathbf{Z}_{4}\times\mathbf{Z}_{18}\times\mathbf{Z}_{15} &\cong \mathbf{Z}_{4}\times\mathbf{Z}_{2}\times\mathbf{Z}_{9}\times\mathbf{Z}_{3}\times\mathbf{Z}_{5} \\ &\cong \mathbf{Z}_{3}\times\mathbf{Z}_{9}\times\mathbf{Z}_{4}\times \dots \end{align*} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.