David Hilbert posed 23 problems he felt would be central to next century of mathematics research. Hilbert's fifth problem concerns the characterization of Lie groups by their actions on topological spaces: to what extent does their topology provide information sufficient to determine their geometry? The "restricted" version of Hilbert's fifth problem (solved by Gleason) asks, more specifically, whether every locally Euclidean topological group is a Lie group. That is, if a group G has the structure of a topological manifold, can that structure be strengthened to a real analytic structure, so that within any neighborhood of an element of G, the group law is defined by a convergent power series, and so that overlapping neighborhoods have compatible power series definitions? Prior to Gleason's work, special cases of the problem had been solved by Luitzen Egbertus Jan Brouwer, John von Neumann, Lev Pontryagin, and Garrett Birkhoff, among others.<ref name="illman">. </ref> Gleason's interest in the fifth problem began in the late 1940s, sparked by a course he took from George Mackey. In 1949 he published a paper introducing the "no small subgroups" property of Lie groups (the existence of a neighborhood of the identity within which no nontrivial subgroup exists) that would eventually be crucial to its solution. His 1952 paper on the subject, together with a paper published concurrently by Deane Montgomery and Leo Zippin, solves affirmatively the restricted version of Hilbert's fifth problem, showing that indeed every locally Euclidean group is a Lie group. Gleason's contribution was to prove that this is true when G has the no small subgroups property; Montgomery and Zippin showed every locally Euclidean group has this property. As Gleason told the story, the key insight of his proof was to apply the fact that monotonic functions are differentiable almost everywhere. On finding the solution, he took a week of leave to write it up, and it was printed in the Annals of Mathematics alongside the paper of Montgomery and Zippin; another paper a year later by Hidehiko Yamabe removed some technical side conditions from Gleason's proof. The "unrestricted" version of Hilbert's fifth problem, closer to Hilbert's original formulation, considers both a locally Euclidean group G and another manifold M on which G has a continuous action. Hilbert asked whether, in this case, M and the action of G could be given a real analytic structure. It was quickly realized that the answer was negative, after