Indecomposable A-module summands in H*V which are unstable algebras
Let p be a prime and denote by A the mod p Steenrod algebra. We determine the indecomposable A-module summands of \(H*(({\mathbb{Z}}/p)^ d;{\mathbb{F}}_ p)\) which admit the structure of an unstable A-algebra. In fact, it turns out that this is equivalent to the problem of determining those indecomposable A-module summands which arise as the mod p cohomology of a space (or even a classifying space of a finite group). We reduce this problem to one in modular representation theory, namely for which d and p is the projective cover of the trivial one dimensional \(GL(d,{\mathbb{F}}_ p)\) representation \({\mathbb{F}}_ p\) a permutation module? Our solution to this latter problem makes use of the classification of subgroups of \(GL(d,{\mathbb{F}}_ p)\) acting transitively on \(({\mathbb{F}}_ p)^ d\setminus \{0\}\) and hence depends on the classification of finite simple groups (on Feit-Thompson's odd order theorem if \(p=2)\).
scientific article; zbMATH DE number 4124602
scientific article