We know how to tell if a topological group is a Lie group: this was famously asked by Hilbert and answered gloriously by Gleason, Montgomery and Zippin in the 50s (a locally compact topological group can be turned into a Lie group iff there is one neighborhood of 1 which does not contain a non-trivial subgroup)

Can one tell when an abstract group is in some way a Lie group?

A natural follow up questions is, of course: if a group is Lie-sable (urgh), is it in a unique way? For comparison, a topological group which can made into a Lie group can be made in exactly one way.

lineari.e. if it is realizable as a subgroup of a matrix group. For this there's the Tits alternative. en.wikipedia.org/wiki/Tits_alternative $\endgroup$ – Ryan Budney May 1 '11 at 20:18