The equivariant $J$–homomorphism for finite groups at certain primes

Abstract

Suppose $G$ is a finite group and $p$ a prime, such that none of the prime divisors of $G$ are congruent to $1$ modulo $p$. We prove an equivariant analogue of Adams’ result that $J^{\prime }=J^{\prime \prime }$. We use this to show that the $G$–connected cover of $Q_G{S}^0$, when completed at $p$, splits up to homotopy as a product, where one of the factors of the splitting contains the image of the classical equivariant $J$–homomorphism on equivariant homotopy groups.