Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory and the theory of topological manifolds. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to smooth manifolds is imposed? The expected answer was in the negative. This was eventually confirmed in the early 1950s. Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.
Classic formulation
A formulation that was accepted for a long period was that the question was to characterize Lie groups as the topological groups that were also topological manifolds. In terms closer to those that Hilbert would have used, near the identity element of the group in question, there is an open set in Euclidean space containing, and on some open subset of there is a continuous mapping that satisfies the group axioms where those are defined. This much is a fragment of a typical locally Euclidean topological group. The problem is then to show that is a smooth function near . Another way to put this is that the possible differentiability class of does not matter: the group axioms collapse the whole gamut.
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: 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 and the connected component of the identity, we have a group extension As a totally disconnected group, has an open compact subgroup, and the pullback of such an open compact subgroup is an open, almost connected subgroup of. In this way, we have a smooth structure on, since it is homeomorphic to, where is a discrete set.
An important condition in the theory is no small subgroups. A topological group, or a partial piece of a group like above, is said to have no small subgroups if there is a neighbourhood of containing no subgroup bigger than For example, the circle group satisfies the condition, while the -adic integers as additive group does not, because will contain the subgroups:, for all large integers. 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 can act faithfully on a closed manifold. Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact groups, as those having no small subgroups.
