 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 . 
