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Is there a name for a subgroup $H < G$ which is not necessarily normal, but such that, for any $f : G \rightarrow Y$ where $Y$ belongs to a particular category (e.g. finite groups, linear groups, etc.), we have $f(H) \trianglelefteq f(G) \qquad$ ?

We could imagine generalizing this concept almost indefinitely: Fix a category $\mathcal{C}$, a subcategory $\mathcal{C}_0$, and a property $P$ enjoyed by certain objects of $\mathcal{C}_0$. We could then consider objects $X$ of $\mathcal{C}_0$ such that, for any object $Y$ of $\mathcal{C}_0$ and epimorphism $f : X \rightarrow Y$, the codomain $Y$ enjoys said property.

If I haven't made a mistake, the situation in the first paragraph could then be obtained by taking $\mathcal{C}$ to be the category of pairs $(H, G)$ with $G$ a group and $H \leq G$ a subgroup, and filling in the remaining conditions in an appropriate way.

Similarly, if we take $\mathcal{C}$ to be the category of topological spaces, $\mathcal{C}_0$ to be the subspaces of $\mathbb{R}$, and $P$ to be boundedness, then we'd get the notion of pseudocompactness.

[ I have no doubt that I'm playing disgustingly fast-and-loose with categorical notions here -- corrections are very welcome. ]

My question: is there (i) a word for the notion in the first paragraph, or (ii) a general nomenclature (along the lines of "pro-X" or "residually-X" or ...) that lets us describe the situation in the second paragraph?

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I don't understand what the relationship between $G, H$ and $X, Y$ are. – Qiaochu Yuan May 5 '12 at 3:05
I don't know of a name, but $H$ has the property if and only if $HN\triangleleft G$ for every normal subgroup $N$ of $G$ with the property that $G/N$ embeds into a $\mathcal{C}$-group. If $\mathcal{C}$ is closed under subgroups and arbitrary products, then there is a unique minimum normal subgroup $N$ of $G$ with this property, and then $H$ has the desired property if and only if $HN$ is normal in $G$. – Arturo Magidin May 5 '12 at 3:11
@Daniel: I think your first paragraph, $f\colon X\to Y$ should be $f\colon G\to Y$. – Arturo Magidin May 5 '12 at 3:11
@Arturo: thanks; fixed. – Daniel McLaury May 5 '12 at 3:54
One example that occurs to me, is the "abelianization of G", that is sending G to G/[G,G], which automatically maps any subgroup H of G to a normal subgroup of G/[G,G]. – David Wheeler May 14 '12 at 15:55

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