The problem
Is every locally compact topological group G that acts effectively on a manifold, or is locally Euclidean, necessarily a Lie group (i.e., carries a real-analytic manifold structure making operations analytic)?
Hilbert's Fifth Problem
historicReferences
History & significance
Hilbert's fifth asked whether continuity assumptions alone force differentiability. Von Neumann (1933) settled the compact case; Pontryagin (1939) the abelian case; Chevalley systematised. The decisive leap came from Andrew Gleason (decomposition via inductiveness) and Deane Montgomery + Leo Zippin (small subgroup normality chains), independently converging in 1952.
The resolution (human proof)
Proven 1952 — Gleason; Montgomery–Zippin (independently, complementary). Corollary: every locally Euclidean topological group is a Lie group, and Hilbert's programme of reducing geometry to group theory gained its foundation. The open frontier moved to NON-Archimedean fields: whether p-adic analogues hold is precisely the territory of modern p-adic Lie theory (Lazard, 1965) — solved there too, but the wild 'no-small-subgroups' characterisation questions continue to shape geometric group theory.