Let $A$ be a finite subset of the group $H$. I am interested in sets with the property that

(1)$\qquad\qquad |\{ab\ \colon\ (a,b)\in A\times A\}| = |A|^{2}$.

Thus $A$ has property (1) if the product set $A^{2}$ is as large as possible.

Is it true that any infinite group has arbitrarily large finite sets satisfying (1)?

I can construct arbitrarily large such sets if $H$ is abelian, or non-Noetherian, or has infinite exponent. So the only case I am not sure about is the case of nonabelian, Noetherian groups with finite exponent.

**EDIT.** As pointed out in the comments, the original question does not make sense for abelian groups. A modified question that does apply to all groups is as follows: can we find sets $A$ and $B$ such that $\min\{|A|,|B|\}$ is arbitrarily large, and $|\{ab\ \colon (a,b)\in A\times B\}|=|A|\cdot|B|$. The comment by @YCor provides us with a method for constructing such sets.