The standard actions of finite groups on spheres are linear actions, or by finite subgroups of an orthogonal group. We prove that, in each dimension d > 5, there exists a finite group which admits a faithful, topological action on the sphere of dimension d but does not admit a faithful, linear actions, i.e. is not isomorphic to a finite subgroup of the orthogonal group O(d+1). The situation remains open for smooth actions.
