In the book Contemporary abstract algebra by Joseph Gallian in page 166 the following things are said :
By using results above in an iterative fashion, one can express the same group (up to isomorphism) in many different forms. For example, we have $$ \mathbb {Z}_2 \oplus \mathbb {Z}_2 \oplus \mathbb {Z}_3 \oplus \mathbb {Z}_5 \approx \mathbb {Z}_2 \oplus \mathbb {Z}_6 \oplus \mathbb {Z}_5 \approx \mathbb {Z}_2 \oplus \mathbb {Z}_{30} \\(*) $$
Similarly, $$ \mathbb {Z}_2 \oplus \mathbb {Z}_2 \oplus \mathbb {Z}_3 \oplus \mathbb {Z}_5 \approx \mathbb {Z}_2 \oplus \mathbb {Z}_6 \oplus \mathbb {Z}_5 \approx \mathbb {Z}_2 \oplus \mathbb {Z}_3 \oplus \mathbb {Z}_2 \oplus \mathbb {Z}_5 \approx \mathbb {Z}_6 \oplus \mathbb {Z}_{10}\\ (**)$$ Thus, $ \mathbb {Z}_2 \oplus \mathbb{Z}_{30} \approx \mathbb{Z}_6 \oplus \mathbb{Z}_{10} $
I am not understanding how they are talking about the isomorphisms between these product groups with the help of the results above. I have tried a lot but can't get through it. Please help. It's really important in my self study.
The results above were :
Theorem 8.2 Let G and H be finite cyclic groups. Then $G \oplus H$ is cyclic if and only if $|G|$ and $|H|$ are relatively prime.
Corollary 1 An external direct product $G_1 \oplus G_2 \oplus ... \oplus G_n$ of a finite number of finite cyclic groups is cyclic iff $|G_i|$ and $|G_j|$ are relatively prime when $i \ne j$.
Corollary 2 Let $ m = n_1n_2 ...n_k $. Then $\mathbb Z_m$ is isomorphic to $\mathbb Z_{n_1} \oplus \mathbb Z_{n_2} \oplus ... \oplus \mathbb Z_{n_k}$ iff $n_i$ and $n_j$ are relatively prime when $i \ne j$.
I think I see it now, as $Z_2 \oplus Z_3 \approx Z_6$ so it is replaced by $Z_6$ and again as $Z_5 \oplus Z_6 \approx Z_{30}$ that's why it is replaced by $Z_{30}$ and doing so on in the next cases too.