The action of the Steenrod algebra on the cohomology of \(p\)-compact groups (Q1011989)

Let \(p\) be a prime, \(A_{p}\) be the mod-\(p\) Steenrod algebra and \(X\) be a \(p\)-compact group, that is a triple \((X, BX, e)\) where: {\parindent=6,5mm \begin{itemize}\item[(1)] \(BX\) is a path connected \(p\)-complete space, \item[(2)] \(e: X \rightarrow \Omega(BX)\) is a homotopy equivalence, \item[(3)] \(H^{*}(X; \mathbb{F}_{p})\) is finite. \end{itemize}} In this paper the authors determine the action of \(A_{p}\), \(p\) an odd prime, on the cohomology of a simply connected \(p\)-compact group such that its integral homology has no \(p\)-torsion. Some interesting examples are given.