Mathematical Proceedings of the Cambridge Philosophical Society


Research Article

On the Morava K-theory of some finite 2-groups


BJÖRN SCHUSTER a1
a1 CRM, Institut d’Estudis Catalans, Apartat 50, E-08193 Bellaterra

Abstract

For any fixed prime p and any non-negative integer n there is a 2(pn [minus sign] 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

(Received December 19 1994)