# Prove that $\mathbb{Z} \times\Bbb Z_2$ and $\mathbb{Z}$ are not isomorphic.

Prove that $$\mathbb{Z} \times\Bbb Z_2$$ and $$\mathbb{Z}$$ are not isomorphic.

Here by $$\mathbb{Z}$$ I mean the group $$(\mathbb{Z},+)$$ and by $$\Bbb Z_2$$ I mean the cyclic group of order 2.

This is exercise 2.3.13 part (a) from Dummit/Foote, Abstract Algebra - I am aware the solution may be posted somewhere online, but I try to look at full solutions as little as possible for my own understanding.

Here is my general argument. Is this a correct approach, and is it missing any important details?

Let $$\Bbb Z_2=\{0,1\}$$. The element $$(0,1)$$ in $$\mathbb{Z} \times\Bbb Z_2$$ has order 2, and since isomorphism preserves order of elements and there exists no integer with order 2, there cannot be an isomorphism between the groups.

• It is correct.${}$ Jan 31 at 4:53
• In a more general context this follows from the uniqueness part of the fundamental theorem on finitely generated abelian groups. Jan 31 at 6:06
• It is correct, but has been discussed often already (see for example here). Jan 31 at 17:36