Is $\mathbb{Z}_{84} \oplus \mathbb{Z}_{72}$ isomorphic to $\mathbb{Z}_{36} \oplus \mathbb{Z}_{168}$?

Solution 1:

They are indeed isomorphic, and to notice this, use the fact that $\mathbb{Z}_{mn} = \mathbb{Z}_n \times \mathbb{Z}_m$ if and only if $\text{gcd}(m,n) = 1$

And since $84 = 12 \cdot 7$ and $72 = 8 \cdot9$ you may decompose the L.H.S into

$\mathbb{Z}_9 \times \mathbb{Z}_8 \times \mathbb{Z}_7 \times \mathbb{Z}_3 \times \mathbb{Z}_4$ and show that something similar may be done with the RHS.

Solution 2:

More generally,

$\Bbb Z_m \times \Bbb Z_n \cong \Bbb Z_{\gcd(m,n)} \times \Bbb Z_{\operatorname{lcm}(m,n)}$

(see this question and this question)

Therefore, $$ \mathbb{Z}_{84} \oplus \mathbb{Z}_{72} \cong \mathbb{Z}_{12} \oplus \mathbb{Z}_{504} \cong \mathbb{Z}_{36} \oplus \mathbb{Z}_{168} $$ because $\gcd(84,72)=12=\gcd(36,108)$ and $\operatorname{lcm}(84,72)=504=\operatorname{lcm}(36,108)$.