If Abelian Group $G \cong G^3$, does it follow $G \cong G^2$?

Question

If an Abelian Group $G$ satisfies $G \cong G^{3}$, does it follow that $G \cong G^2$ ?

It seems elementary but I can't find it on a standard textbook exercise, and maybe simply because it's false.

sorry, Edit: I mistyped I mean $G\cong G^3$ implies $G \cong G^2$.

Before edited it was the converse, which is of course true (see Hagen von Eitzen's answer)

Answer 1

You have $G\cong G^2=G\times G\cong G\times G^2=G^3$ (abelian or not, in any category with finite products)