# Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group, and wants to be a sort of "converse".

Here I am taking an abstract group $G$ and looking for necessary conditions for it to admit the structure of a Lie group.

Of course I am thinking of structures with underlying manifold not $0$-dimensional (since every group can be given the discrete topology this wouldn't be interesting).

In the mentioned question they take a manifold, put a group structure on it and look at the constraints this imposes on its "geometric structure" (homotopy, co-homology, bundles...); so here I want to take a group, put a manifold structure over it and find constraints on its "algebraic structure" (subgroups, quotients, representation, whatever...).

In this question: Given a group $G$, how many topological/Lie group structures does $G$ have? there is a wonderful answer which addresses the problem of existence and uniqueness of Lie structure on a topological group.

But this is not what I'm looking for, because there it is assumed a priori that $G$ admits the structure of a topological group; then this structure is "smoothened" to obtain a Lie structure. What I want to do is to start with a $G$ without any topological structure. So my question could also be rephrased as:

Which algebraic properties distinguish Lie groups from abstract groups?

To be explicit, the no-small-subgroups property quoted in the second question is the kind of things I'm looking for.

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