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I'm not asking for a proof that every simple abelian group is simple; that's a fairly trivial question. To lead into my real question, I started thinking about this question:

Let $\cal C$ be a category with a terminal object, and let $\cal B$ be a full subcategory which contains a terminal object from $\cal C$ (we can assume $\cal B$ is isomorphism closed). If $X$ is an object of $\cal B$ which is simple in $\cal B$, does it follow that $X$ is a simple object of $\cal C$?

The answer to this question is false by considering the category

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and considering the subcategory

$\hskip2in$enter image description here

In these examples, $X$ is simple in the latter but not the former. This made me wonder

What is special about the relationship between Grp and Ab that implies every simple object of Ab is simple in Grp?

I thought it might be because the inclusion functor from Ab to Grp has a left adjoint (the abelianization functor), but this is also true in the categories above (collapse $Y$ to $X$).

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$\mathsf{Ab}$ is closed under quotients taken in $\mathsf{Grp}$.

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