Hilbert's Fifth Problem - No Small Subgroups

An important condition in the theory is no small subgroups. A topological group G, or a partial piece of a group like F above, is said to have no small subgroups if there is a neighbourhood N of e containing no subgroup bigger than {e}. For example the circle group satisfies the condition, while the p-adic integers Z_p as additive group does not, because N will contain the subgroups

for all large integers k. This gives an idea of what the difficulty is like in the problem. In the Hilbert–Smith conjecture case it is a matter of a known reduction to whether Z_p can act faithfully on a closed manifold. Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact groups, as those having no small subgroups.