The splitting of cohomology of \(p\)-groups with rank 2 (Q1689403)

Let \(p\) be an odd prime and \(BP\) the classifying space of a \(p\)-group \(P\) with \(\text{rank}_p(P)=2\). First, the authors continue their work from [J. Algebra 409, 265--319 (2014; Zbl 1309.55004); ibid. 451, 461--493 (2016; Zbl 1333.55007)], the study of the stable splitting \(BP\simeq X_1\vee\cdots\vee X_n\) and the splitting of cohomology \(H^\ast(P)\approx H^\ast(X_1)\oplus\cdots\oplus H^\ast(X_n)\) for \(\ast >0\), where \(X_i\) are irreducible spaces in the stable homotopy category for \(i=1,\ldots,n\). Using the stable splitting of \(BP\), the decomposition of \(H^\ast(P)=H^\ast(P;\mathbb{Z})/(p,\sqrt{0})\) and \(H^{even}(P) = H^{even}(P;\mathbb{Z})/p\) are presented. At the end, relations for these stable splittings to Chow rings and motives are added.