An Artin Relation (Mod 2) for Finite Group Actions on Spheres

Recently it has been shown that whenever a finite group G (not a /7-group) acts on a homotopy sphere there is no general numerical relation which holds between the various formal dimensions of the fixed sets of ^-subgroups (p dividing the order of (7). However, if G is dihedral of order 2q (q an odd prime power) there is a numerical relation which holds (mod 2). In this paper, actions of groups G which are extensions of an odd order /?-group by a cyclic 2-group are considered and a numerical relation (mod 2) is found to be satisfied (for such groups acting on spheres) between the various dimensions of fixed sets of certain sub groups; this relation generalises the classical Artin relation for dihedral actions on spheres.