# Prove that these two products of cyclic groups are isomorphic to each other.

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.

• – St Vincent Jun 10 '15 at 7:29

\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*}