# Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies

Let $N\triangleleft G$. Then $G$ is solvable if and only if both $N$ and $G/N$ are solvable.

This says solvability is a quotient-lifting property in the sense that it can be passed down to substructures and quotients, and it can also be lifted up.

I think it is natural to abstract this a little bit. Say we are studying some structures $\mathcal{A}$, with a substructure $J$ and the quotient $\mathcal{A}/J$, and P is a certain property. We say P is quotient-lifting if

$\mathcal{A}$ has P if and only if both $J$ and $\mathcal{A}/J$ have P.

Apart from solvability of groups, being Noetherian is a quotient-lifting property of modules.

Quotient-lifting properties allow us to break a large structure to smaller parts, and probably would make life easier.

Also, whenever we have a quotient-lifting property P, it allows us to define a maximal object with property P inside a given $\mathcal{A}$. This follows from similar argument in

Let $N,H\triangleleft G$. If both $N$ and $H$ are solvable then $NH$ is solvable. (So in particular, we can take all solvable normal subgroup $\{N_{\alpha}\}$ in $G$ then $\Pi N_{\alpha}$ is the maximal solvable normal subgroup in $G$.)

I hope to learn more about these properties: 1) do we have some general results concerning quotient-lifting properties? 2) what are other nice examples of quotient-lifting properties? 3) apart from abstract algebra, do we have such things in other parts of mathematics? (Analysis? Geometry? )

Thanks!

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