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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.
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