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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.

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    $\begingroup$ There are situations where "mono" and "injective" are not the same (though they are the same in groups). $\endgroup$
    – Randall
    Mar 2, 2021 at 19:29
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    $\begingroup$ "injective" has to do with sets and points. "monomorphism" is a universal mapping property, can be stated without mentioning the elements of the group. In some categories they are different. If they are the same for groups, it is something that must be proved. $\endgroup$
    – GEdgar
    Mar 2, 2021 at 19:30
  • $\begingroup$ See this question and answer for an example of a subclass of groups (the divisible groups) where monomorphisms need not be injective. Basically, "monomorphism" is a categorical context, "injective" is a set-theoretic concept, and the author is showing they coincide in the class of all groups. Just like we also establish the equivalence between "isomorphism" (has an inverse) and "bijective homomorphism" for groups/vector spaces. (But not equivalent for topological spaces!). $\endgroup$ Mar 3, 2021 at 0:06

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