Abstract: We are interested in classifying all finite groups which can act on a finite $CW$-complex homotopy equivalent to a sphere, such that all isotropy subgroups are rank one groups, meaning that they do not include $Z/p\times Z/p$ for any prime $p$. The similar question for free actions (all isotropy subgroups are trivial) has been answered completely by the works of P.A. Smith and R. Swan. There is a complete list of such groups (they are finite groups with periodic group cohomology) and we would like to obtain a similar list for actions with rank one isotropy. This is joint work with Ian Hambleton.