I have a group $G$, and for all $g \in G$ and $a,b \in \mathbb{Z}$ it makes sense to talk about the element $g^{a \over b} \in G$.
To get some intuition, I've been thinking about what it would mean to do this to the integers over addition: it's not a group unless we extend the set to all rational numbers. Likewise if we consider the rationals over multiplication, it's not a group unless we extend the set to all real numbers. But I would like to see some formal treatments of this structure and am wondering where to look.
Is it true that this could be an alternative definition for Abelian groups? or is this something else entirely?