How to show that $Z/12Z×Z/90Z×Z/25Z$ and $Z/100Z×Z/30Z×Z/9Z$ are isomorph? How to show that $G = \Bbb Z/12\Bbb Z \times \Bbb Z/90 \Bbb Z\times \Bbb Z/25 \Bbb Z$ and  $H = \Bbb Z/100 \Bbb Z \times \Bbb Z/30\Bbb Z\times \Bbb Z/9\Bbb Z$ are isomorph?
The way I would go is to use the decomposition in primary groups:
For G:


*

*$12 = 2^2 * 3$

*$90 = 3^2*2*5$

*$25 = 5^2$


So $G \simeq G(2) \times G(3) \times G(5)$ with


*

*$G(2)= (\Bbb Z/2\Bbb Z) \times (\Bbb Z/2^2\Bbb Z)$

*$G(3)= (\Bbb Z/3\Bbb Z) \times (\Bbb Z/3^2\Bbb Z)$

*$G(5)= (\Bbb Z/5\Bbb Z) \times (\Bbb Z/5^2\Bbb Z)$


For H:


*

*$100 = 5^2 * 2^2$

*$30 = 5*3*2$

*$9 = 3^2$


So $H \simeq H(2) \times H(3) \times H(5)$ with


*

*$H(2)= (\Bbb Z/2\Bbb Z) \times (\Bbb Z/2^2\Bbb Z)$

*$H(3)= (\Bbb Z/3\Bbb Z) \times (\Bbb Z/3^2\Bbb Z)$

*$H(5)= (\Bbb Z/5\Bbb Z) \times (\Bbb Z/5^2\Bbb Z)$


As $G$ and $H$ have the same decomposition they are isomorph.
Is it right?
Is there a better/faster/easier way to prove it?
 A: Your proof using the primary decomposition is fine and systematic.
You can also use the Chinese remainder theorem to recombine the factors into the invariant factor decomposition:
$
G
= \Bbb Z/12\Bbb Z \times \Bbb Z/90 \Bbb Z\times \Bbb Z/25 \Bbb Z
\cong \Bbb Z/3\Bbb Z \times \Bbb Z/4\Bbb Z \times \Bbb Z/9 \Bbb Z \times \Bbb Z/10 \Bbb Z\times \Bbb Z/25 \Bbb Z
\cong \Bbb Z/30\Bbb Z\times \Bbb Z/900 \Bbb Z
$
$
H 
= \Bbb Z/100 \Bbb Z \times \Bbb Z/30\Bbb Z\times \Bbb Z/9\Bbb Z
\cong \Bbb Z/30\Bbb Z\times \Bbb Z/100 \Bbb Z \times \Bbb Z/9\Bbb Z
\cong \Bbb Z/30\Bbb Z\times \Bbb Z/900 \Bbb Z
$
A: I would have displayed it this way.
\begin{align}
   \mathbb Z_{12} × \mathbb Z_{90} × \mathbb Z_{25}
   &\cong (\mathbb Z_4 \times \mathbb Z_3) \times
         (\mathbb Z_2 \times \mathbb Z_9 \times \mathbb Z_5) \times
         (\mathbb Z_{25}) \\
   &\cong (\mathbb Z_4 \times \mathbb Z_{25}) \times
         (\mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5) \times
         (\mathbb Z_9) \\
   &\cong \mathbb Z_{100} × \mathbb Z_{30} × \mathbb Z_9
\end{align}
