© 2014 Mathematical Sciences Publishers. All Rights reserved. A p–local compact group consists of a discrete p–toral group S, together with a fusion system and a linking system over S which define a classifying space having very nice homotopy properties. We prove here that if some finite regular cover of a space Y is the classifying space of a p–local compact group, then so is Y^p. Together with earlier results by Dwyer and Wilkerson and by the authors, this implies as a special case that a finite loop space determines a p–local compact group at each prime p.