The authors of this paper classify the locally finite groups in which every element has prime-power order. The finite such groups were completely classified between 1957 and 1994, with many authors contributing. The history of this is given in detail by the authors.
A major step in the finite case was Suzuki’s 1962 classification of the finite simple such groups. There are only eight such finite simple groups and in view of this it is not surprising that there are no infinite, locally finite, simple such groups. In general, the authors’ classification theorem of the locally finite such groups closely mirrors the finite case.