The definition my teacher gave in his course are:
-Let $G$ and $H$ be groups, let $f:G \longrightarrow H$ an homomorphism of groups. We call $f$ a group monomorphism if for all group $K$ and any pair of homomorphisms $\alpha: K \longrightarrow G$ and $\beta: K \longrightarrow G$, the equality $f\alpha=f\beta$ implies $\alpha=\beta$.
Then he proves the following:
Let $G$ and $H$ be groups, $f: G \longrightarrow H$ an homomorphism of groups. The statements
a) $f$ is injective;
b) $f$ is a monomorphism;
c) $Ker(f)=\{e_{G}\}$
are equivalent.
He proceeds proving the implications a) $\implies$ b), b) $\implies$ c), c) $\implies$ d), then he does not prove directly the equivalence between a) and b), but I'm aware that in general given any function $f:G \longrightarrow H$ (where G and H are any sets), $f$ being injective amounts to being "left-cancellable" (i.e. if $\alpha: K \longrightarrow G$ and $\beta: K \longrightarrow G$ are such that $f\alpha=f\beta$, then $\alpha=\beta$), then I don't know if starting defining a monomorphism as above and then proving the equivalence of being a monomorphims and being injective is redundant or if there's something important behind it I can't see.
