A group is an invertible monoid.
A group is uniquely invertible.
Proof.Let $(G,*,e)$ be a group and $x\in G$. If both $y$ and $y’$ are inverses of $x$,
\[y = ye = y(xy') = (yx)y' = ey' = y'.\]
In an abelian group $(G,+)$, we write $-a$ for the inverse of $a$ and
\[a-b = a+(-b).\]
A group is cancellative.
A monoid becomes a group as soon as it is left-invertible or right-invertible.
Proof.By PROP-MO-LR.
Let $G$ be a group.
For any $a\in G$,
\[(a^{-1})^{-1} = a.\]For any $a$, $b\in G$,
\[(ab)^{-1} = b^{-1}a^{-1}.\]
If $M$ is a monoid, its invertible subset $\inv M$ is a group.