Brown-Peterson cohomology from Morava E-theory

We prove that the -completed Brown-Peterson spectrum is a retract of a product of Morava -theory spectra. As a consequence, we generalize results of Kashiwabara and of Ravenel, Wilson and Yagita from spaces to spectra and deduce that the notion of a good group is determined by Brown-Peterson cohomology. Furthermore, we show that rational factorizations of the Morava -theory of certain finite groups hold integrally up to bounded torsion with height-independent exponent, thereby lifting these factorizations to the rationalized Brown-Peterson cohomology of such groups.