Morava K-theory of classifying spaces: Some calculations

We develop some basic methods for calculating Morava K-theories of compact Lie groups, and compute certain pivotal examples. We show that K̃(2)∗BP has odd elements, where P is the 3-Sylow subgroup of GL4(Z/3). This disproves a conjecture of Hopkins, Kuhn and Ravenel. We also calculate Morava K-theories of semidirect products of cyclic groups with elementary abelian groups, and prove a new theorem on complex-oriented cohomology of BO(K).