Hilbert's Fifth Problem - Solution

Solution

The first major result was that of John von Neumann in 1933, for compact groups. The locally compact abelian group case was solved in 1934 by Lev Pontryagin. The final resolution, at least in this interpretation of what Hilbert meant, came with the work of Andrew Gleason, Deane Montgomery and Leo Zippin in the 1950s.

In 1953, Hidehiko Yamabe obtained the final answer to Hilbert’s Fifth Problem: a connected locally compact group G is a projective limit of a sequence of Lie groups, and if G "has no small subgroups" (a condition defined below), then G is a Lie group. However, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers.

More generally, every locally compact, almost connected group is the projective limit of a Lie group. If we consider a general locally compact group G and the connected component of the identity G₀, we have a group extension

As a totally disconnected group G/G₀ has an open compact subgroup, and the pullback G′ of such an open compact subgroup is an open, almost connected subgroup of G. In this way, we have a smooth structure on G, since it is homeomorphic to G′ × G′ / G₀, where G′ / G₀ is a discrete set.