# When is a monoid contained in a group?

As stated in the answer of Is the forgetful functor from groups to monoids right adjoint? , the forgetful functor $U:\mathbf{Grp}\rightarrow\mathbf{Mon}$ has a left adjoint $G$, and Grothendieck's construction is involved.

Now, my intuition tells me that for a monoid $M$ the following holds:

$M$ can be embedded in some group if and only if $M \longrightarrow G(M)$ is injective.

Is this true? Can we find a characterization of such monoids?

• For commutative or finite monoids, cancellation is a sufficient condition. – Slade May 7 '15 at 18:27

Your statement is correct. The universal property of $G(M)$ is that a homomorphism of monoids $M \to G$, where $G$ is a group, extends uniquely to a group homomorphism $G(M)\to G$ via the map $M\to G(M)$.
It follows that if there exists an injective homomorphism $M\to G$, then the map $M\to G(M)$ is injective.