This question is motivated by a previous one:
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:
there is a wonderful answer which adresses 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.