Groupoid embedding into a group

Question

A necessary condition for a groupoid $(G,∗)$ to embed into a group is that for all $a,b\in G$ then $a ∗ a^{-1}=b∗b^{−1}$.

Question: Is this necessary condition also sufficient?

This question came to my mind after reading this post 2 days before.

Answer 1

I would say so. Let's say a group is a one-object categorical groupoid. Under the equivalence of definitions described in the Wikipedia article an algebraic groupoid with the property $a\ast a^{-1}=b\ast b^{-1}$ for all $a,b$ becomes a one-object groupoid, hence a group.