Editing Hilbert's fifth problem (section)

{{Short description|Problem in Lie group theory}}
'''Hilbert's fifth problem''' is the fifth mathematical problem from the [[Hilbert problems|problem list]] publicized in 1900 by mathematician [[David Hilbert]], and concerns the characterization of [[Lie group]]s.

The theory of Lie groups describes [[continuous symmetry]] in mathematics; its importance there and in [[theoretical physics]] (for example [[quark theory]]) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of [[group theory]] and the theory of [[topological manifold]]s. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to [[smooth manifold]]s is imposed?

The expected answer was in the negative (the [[classical group]]s, the most central examples in Lie group theory, are smooth manifolds). 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.