  We prove that the $p$-completed Brown-Peterson spectrum is a retract of a
product of Morava $E$-theory spectra. As a consequence, we generalize results
of Ravenel-Wilson-Yagita and Kashiwabara from spaces to spectra and deduce that
the notion of good group is determined by Brown-Peterson cohomology.
Furthermore, we show that rational factorizations of the Morava $E$-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.
