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I would like to solve the Problem 2.12 in Szekeres, A Course in Modern Mathematical Physics:

Show that the class of groups as objects with homomorphisms between groups as morphisms forms a category – the category of groups (see Section 1.7). What are the monomorphisms, epimorphisms and isomorphisms of this category?

In the first part, I consider category $\mathbf{Grp}$ as the class of all groups and the group homomorphism as category morphism, and I show the needed properties.

The confusing part is to find the monomorphisms, epimorphisms and isomorphisms of this category.

In https://en.wikipedia.org/wiki/Group_homomorphism we have the associations of monomorphism with injective, epimorphism with surjective, and isomorphism with bijective. But in https://en.wikipedia.org/wiki/Homomorphism#Category_theory in paragraph in about Category theory I read:

However, the definitions in category theory are somewhat different.

I interpret the sentence to means: Category monomorphism, epimorphism, and isomorphism are different than the usual homomorphism definitions (for rings, groups, aso) Is this true for groups?

Concretely I have found no way to derive, for instance, that monomorphisms from the group category are injective homomorphism. I mean: starting from:

The monomorphisms of $\mathbf{Grp}$ are the homomorphisms $\phi:G\to H$ such that

$\forall X\in\mathbf{Grp},\forall \alpha,\alpha':X\to G$, we have: $\phi\circ\alpha=\phi\circ\alpha'\implies \alpha=\alpha'$

And get to $\phi(x)=\phi(y)\implies x=y$.

Another observation. In Szekeres the Problem is stated after very few definitions (and no theorem!). Thus I cannot use things like functors as in Is every monomorphism an injection?

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It is obvious that injective (respectively, surjective) group homomorphisms are monomorphisms (respectively, epimorphisms).

Monomorphisms in the category of groups are injective maps. Indeed, suppose $\phi\colon G\to H$ is a monomorphism; consider $\alpha\colon\ker\phi\to G$, the canonical injection, and $\beta\colon\ker\phi\to G$, $\beta(x)=1$. Then $\phi\circ\alpha=\phi\circ\beta$: what does $\alpha=\beta$ entail?

Epimorphisms in the category of groups are surjective, but this is a bit more difficult to show (one needs to define an action on the set of cosets of $H$ by the image of $\phi$).

The standard example of a nonsurjective epimorphism in a category is the embedding $\mathbb{Z}\to\mathbb{Q}$ in the category of rings, which is both a monomorphism (obvious) and an epimorphism (try it).

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  • $\begingroup$ A proof of the fact that a group epi is surjective is to be found in Carl Linderholm, "A Group Epimorphism is Surjective", American Mathematical Monthly 77, pp. 176-177. From the article (paraphrased): Of the 6 assertions (mono iff injective, epi iff surjective, iso iff bijective) the only one that is not "very easy" is epi $\implies$ surjective. $\endgroup$ – PeptideChain Jul 14 '16 at 18:43

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