IndecomposableA-module summands inH * V which are unstable algebras

Letp be a prime and denote byA the modp Steenrod algebra. We determine the indecomposableA-module summands ofH*((ℤ/p)) ^d ;F _p which admit the structure of an unstableA-algebra. In fact, it turns out that this is equivalent to the problem of determining those indecomposableA-module summands which arise as the modp 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 whichd andp is the projective cover of the trivial one dimensional GL(d,F _p ) representationF _p a permutation module. Our solution of this latter problem makes use of the classification of subgroups of GL(d,F _p ) acting transitively on (F _p ) ^d \{0} and hence depends on the classification of finite simple groups (on Feit-Thompson's odd order theorem ifp=2).