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.

  • 11
    $\begingroup$ It is correct.${}$ $\endgroup$
    – markvs
    Jan 31 at 4:53
  • 3
    $\begingroup$ In a more general context this follows from the uniqueness part of the fundamental theorem on finitely generated abelian groups. $\endgroup$ Jan 31 at 6:06
  • $\begingroup$ It is correct, but has been discussed often already (see for example here). $\endgroup$ Jan 31 at 17:36
  • $\begingroup$ See also the solutions to homework. $\endgroup$ Jan 31 at 17:41
  • $\begingroup$ Indeed, that is why I added my remark about desiring not to look at other individuals' full solutions wherever possible. $\endgroup$ Jan 31 at 18:30


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