How to construct an isomorphism?

Question

I know the group $\mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_6$ (since $\mathbb{Z}_3 \times \mathbb{Z}_2$ is isomorphic to $\mathbb{Z}_6$ since $2$ and $3$ are coprimes).

But how would I construct such an ismorphism?

You can use the Chinese Remainder Theorem. – Viktor Vaughn Oct 19 '17 at 22:29 — Viktor Vaughn, Oct 19 '17 at 22:29

score 1 · Accepted Answer · answered Oct 19 '17 at 22:36

1

You can make it explicit by projecting the elements of $Z/6$ to the respective elements in $Z/3$ and $Z/2$, i.e. $$Z/6 \rightarrow Z/3 \times Z/2, i \mapsto (i \text{ mod } 3, i \text{ mod }2).$$ As you say, 2 and 3 are coprime and thus the Chinese remainder theorem tells you that this is an isomorphism.

answered Oct 19 '17 at 22:36

user028230

131

The isomorphism is supposed to be between $Z_3 \times Z_3 \times Z_2$ and $Z_3 \times Z_6$ – Dionel Jaime Oct 19 '17 at 22:45
2

I think it is pretty obvious that you take the identity on the first component. Or am I overseeing something? – user028230 Oct 19 '17 at 22:48

Dionel Jaime · Answer 2 · 2017-10-19T22:47:14.267

0

An isomorphism $f: \mathbb{Z_3 \times Z_2} \to \mathbb{Z_6}$ can be determined by $f(1,1) \to 1$

So it would make sense that the map $\mathbb{Z_3 \times Z_3 \times Z_2} \to \mathbb{Z_3 \times Z_6}$ could be given by

$$(0,a,b) \mapsto (0,f(a,b)) \\(1,a,b) \mapsto (1,f(a,b))\\(2,a,b) \mapsto(2,f(a,b)) $$

edited Oct 19 '17 at 22:47

answered Oct 19 '17 at 22:32

Dionel Jaime

3,908

and a ∈ Z3 and b ∈ z2? – JPRP Oct 19 '17 at 22:37
$a \in Z_3$ and $b \in Z_2$ – Dionel Jaime Oct 19 '17 at 22:38
Yes, this is clearly a bijection. Now just check that it's a homomorphism. – Dionel Jaime Oct 19 '17 at 22:41

How to construct an isomorphism?

2 Answers2

Linked