On stably decomposing products of classifying spaces (Q5926362)

Let \(G= P \times Q\) be a product of two finite \(p\)-groups, \(p\) a prime. The classifying space \(BG\) is called smash decomposable if the indecomposable summands of \(\Sigma^\infty BG_+\) are given by simply smashing together those of \(\Sigma^\infty BP_+\) and \(\Sigma^\infty BH_+\). In the simplest case where \(P\) and \(Q\) are abelian \(p\)-groups \(BG\) is smash decomposable if and only if \(P\) and \(Q\) have no cyclic factors of equal order, which is a result due to \textit{J. C. Harris} and \textit{N. J. Kuhn} [Math. Proc. Camb. Philos. Soc. 103, No. 3, 427-449 (1988; Zbl 0686.55007)]. In the present article the authors generalize this result to the case where just one of the factors of \(G\) is abelian. Moreover, they also give some conditions that imply smash decomposability of \(BG\) in cases where neither \(P\) nor \(Q\) is abelian. The results are used to verify smash decomposability for a variety of examples. The list of examples includes \(Q_{2^m}\times {\mathbb Z}/2^n\) and \(D_{2^m}\times {\mathbb Z}/2^n\) in case \(m>2\) and \(n>1\), as well as \(Q_8 \times Q_8\) and \(D_8 \times Q_8\) and others.