Isomorphism between finite groups

Question

What can be the general statements made, when two finite groups can be isomorphic to each other: $\mathbb{Z}_{pq}$ and $\mathbb{Z}_{p} \times \mathbb{Z}_{q}$

Say $$\mathbb{Z}_{pq} \simeq\mathbb{Z}_{p} \times \mathbb{Z}_{q},$$

what are the conditions for $p$ and $q$ and their factorization relations?

How do we show this in the most elegant way? And the intuitive way?

Possible duplicate of https://math.stackexchange.com/questions/1490608/is-there-a-slick-way-to-test-whether-bbb-z-mn-cong-bbb-z-m-oplus-bbb-z-n — lhf, Apr 10 '18 at 00:23
Possible duplicate of Is there a slick way to test whether $\Bbb Z_{mn}\cong \Bbb Z_m\oplus \Bbb Z_n$? — KonKan, Apr 10 '18 at 02:22

score 2 · Accepted Answer · answered Apr 10 '18 at 00:25

2

The general result is

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

(see this question and this question)

In particular,

$\Bbb Z_m \times \Bbb Z_n \cong \Bbb Z_{mn} $ when $\gcd(m,n)=1$.

answered Apr 10 '18 at 00:25

lhf

216,483

Adapted from https://math.stackexchange.com/a/2483130/589 – lhf Apr 10 '18 at 00:25
how do we know this is the most general result? thanks +1. – annie marie cœur Apr 10 '18 at 00:45
@annieheart, there is almost never the most general result. In the context of your question, a major general result is the fundamental theorem of finitely generated abelian groups. – lhf Apr 10 '18 at 00:47

Isomorphism between finite groups

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