Prove that $\Bbb{Z}×\Bbb{Z}$ and $\Bbb{Z}×\Bbb{Z}×\Bbb{Z}$ is not isomorphic groups.

Question

Considering them as rings, I can prove that they are not isomorphic as they have different nos of idempotent elements , but in case of group isomorphism I am unable to prove it.

Answer 1

Hint:

$\mathbb Z\times\mathbb Z$ can be generated by $2$ elements.

$\mathbb Z\times\mathbb Z\times\mathbb Z$ cannot be generated by $2$ elements.

(Prove that for two elements $(a,b,c)$ and $(u,v,w)$ the set $\{n(a,b,c)+m(u,v,w)\mid n,m\in\mathbb Z\}$ is Always a proper subset of $\mathbb Z\times\mathbb Z\times\mathbb Z$)

Answer 2

That results from the Invariant Basis Number property in commutative rings: they're frr $\bf Z$-modules, with a basis of $2$ and $3$ vectors respectively.

Answer 3

Hint: consider the possible cardinalities of sets of odd elements such that the sum of any two is odd. Alternatively, show that if $G_1$, $G_2$ are isomorphic abelian groups, then $G_1/2G_1$ and $G_2/2G_2$ are isomorphic.