Generators of the cohomology of M(n) as a module over the odd primary Steenrod algebra

Given an odd prime p, we determine a minimal generating set for the mod p cohomology H*(M(n)) as a module over the mod p Steenrod algebra A(p), where M(n) is a stable summand of the classifying space B(Z/p)n induced by the Steinberg idempotent. Actually, we describe a basis for the quotient graded vector space of H*(M(n)) modulo all decomposable elements, which lifts to a minimal generating set for H* (M(n)) as an A(p)-module.