Groups Without Small Subgroups

Abstract

A topological group is said to have small subgroups if every neighborhood of the identity contains an entire subgroup of more than one element. It is easily deduced from the structure of the one-parameter subgroups that a Lie group does not have small subgroups. The possible existence of small subgroups in a locally euclidean group has long been recognized as one of the chief difficulties involved in Hilbert's fifth problem. Our objective is to prove that every finite dimensional locally compact group which has no small subgroups is a Lie group. This partial solution of Hilbert's fifth problem offers a new non-analytic criterion sufficent to insure that a group has an analytic structure. As such it represents a considerable step in the direction of ultimate solution of the problem. In fact, Montgomery and Zippin, using the present result inductively, have succeeded in proving the non-existence of small subgroups in locally connected groups of finite dimension. (See the following paper.) Together these results constitute an affirmative solution to Hilbert's fifth problem. The proof depends on the construction of a suitable one-parameter semi-group of compact sets, a technique initiated by the author [3] to prove the existence of arcs. It seems unlikely that the strength of the method is now exhausted; on the contrary, there is definite hope of proving the further conjecture on the structure of locally compact groups: That every connected locally compact group is a projective limit of Lie groups.