New results and old problems in finite transformation groups

1. Definitions. A transformation group (G, X) consists of a group G acting on a topological space X to form a group of homeomorphisms of X onto itself. I t will be understood throughout this paper that G is finite. For a given transformation group or "action" (G, X) and subset HQGy we denote by F(H\ G, X) the fixed-point set of H—that is, the points x such that hx = x for h gx are of class C. When i = 0 we shall drop the manifold condition on X; every action is then of class C°. A differentiable action is a enaction. (G, X) is orthogonal if X is a euclidean sphere or an open submanifold of a euclidean space and the trans