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Here is a question I came across recently:

If a morphsim in Grp (category of groups) is an epimorphism, then it is a surjective group homomorphsim.

I believe it boils down to show that for any group $G$ and its proper subgroup $H$, there exists a group $G'$ and two group homomorphisms $\phi, \psi: G \to G'$ such that $\phi|_{H} = \psi|_{H}$ but $\phi \neq \psi$. However, I am stuck at this point.

There are also two mirror questions related to this one:

If a morphsim in R-Mod (here $R$ is assumed to be unital) is an epimorphism, then it is a surjective module homomorphsim;

If a morphsim in Ring (category of rings) is an epimorphism, then it is NOT necessarily a surjective ring homomorphsim.

The first I don't know how to prove either while for the second question I found the example of the embedding $\mathbb{Z} \hookrightarrow \mathbb{Q}$. It seems weird to me that between rings there is a distinction between epimophisms and surjective maps while in groups or modules there isn't. Are there some deep reasons behind this?

JNF
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    Does this post clarify it a bit? – Dietrich Burde Sep 25 '23 at 12:49
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    There are varieties of groups where there are epimorphisms that are not surjective; on the other hand, it is easy to find epimorphisms of semigroups that are not surjective. The embedding of $\mathbb{Z}$ into $\mathbb{Q}$ is such an example (looking at the multiplicative semigroup structure). To prove epimorphisms in $R$-Mod are surjective, just consider the canonical map into the cokernel and the zero map nto the cokernel. – Arturo Magidin Sep 25 '23 at 17:06
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    See also here, here, and here. This one also has a lot of good references. – Arturo Magidin Sep 25 '23 at 17:08

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