Editing Hilbert's fifth problem (section)

== No small subgroups ==
An important condition in the theory is '''[[no small subgroup]]s'''.  A topological group {{math|''G''}}, or a partial piece of a group like {{math|''F''}} above, is said to have ''no small subgroups'' if there is a neighbourhood {{math|''N''}} of {{math|''e''}} containing no subgroup bigger than {{math|{{mset|''e''}}}}. For example, the [[circle group]] satisfies the condition, while the [[p-adic integers|{{math|''p''}}-adic integers]] {{math|'''Z'''<sub>''p''</sub>}} as [[Abelian group|additive group]] does not, because {{math|''N''}} will contain the subgroups: {{math|''p<sup>k</sup>''&thinsp;'''Z'''<sub>''p''</sub>}}, for all large integers {{math|''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 {{math|'''Z'''<sub>''p''</sub>}} can act faithfully on a [[closed manifold]]. Gleason, Montgomery and Zippin characterized Lie groups amongst [[locally compact group]]s, as those having no small subgroups.