Higher homotopy associativity of power maps on finite $H$ -spaces

Let $p$ be an odd prime, and let $\lambda\in\mathbb{Z}$ . Consider the loop space $Y_{t}=S^{2t-1}_{(p)}$ for $t\ge1$ with $t|(p-1)$ . Then we first determine the condition for the power map $\varPhi_{\lambda}$ on $Y_{t}$ to be an $A_{p}$ -map. We next assume that $X$ is a simply connected $\mathbb{F}_{p}$ -finite $A_{p}$ -space and that $\lambda$ is a primitive $(p-1)$ st root of unity mod $p$ . Our results show that if the reduced power operations $\{\mathscr{P}^{i}\}_{i\ge1}$ act trivially on the indecomposable module $QH^{*}(X;\mathbb{F}_{p})$ and the power map $\varPhi_{\lambda}$ on $X$ is an $A_{n}$ -map with $n\gt (p-1)/2$ , then $X$ is $\mathbb{F}_{p}$ -acyclic.