For any fixed prime p and any non-negative integer n
there is a 2(pn − 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.
Bruner–Greenlees conjecture on real connective K theory of generalized quaternion groups