To What Extent Does the Cartesian Product for Algebraic Structures Generalize?

To What Extent Does the Cartesian Product for Algebraic Structures
Generalize?

I admit this question is quite general.
If we have a group (or perhaps some other algebraic structure) $G$, we can
define the Cartesian product $G\times G$ of $G$ with itself. And then
powers of $G$ as $G^{\times m} = G \times G~ \times ...~m$
times$~...\times ~ G$.
In the case above $m$ is required to be a natural number. My question is
when are we allowed to let $m$ be negative or rational. In the case of
finite Abelian Groups we can obviously factor the cyclic components of a
group as we would an integer and hence $(\mathbb{Z}_p \times
\mathbb{Z}_p)^{\times 1/2} = \mathbb{Z}_p$ et cetera. But I have no idea
what the requirements would be for infinite or non Abelian groups.
This still doesn't address the possibility of negative $m$. I haven't been
able to think of anything on this subject, other than how $G^{\times -1}$
shouldn't have to be a group, like how $\frac{1}{n}$ isn't always an
integer where $n \in \mathbb{Z}$. It would actually be rather strange if
it was a group, because then we would have to worry about how many
elements it has. But if not a group, what could it be. Something
devilishly complicated no doubt!