A criterion for delooping the fibre of the self-map of a sphere with degree a power of a prime (Q1066513)

Let F be the homotopy fibre of the self-map of the (2n-1)-dimensional sphere of degree \(p^ j\), where p is an odd prime. In Mem. Am. Math. Soc. 268 (1982; Zbl 0487.20029), the second author showed that for certain values of p, n and j, the fibre F deloops. In all these cases n divides p-1. In this paper the authors show that F is a loop space if and only if n divides p-1. In the first part of the paper two methods are given for constructing deloopings. One of these deloopings is of the form \((BG^+)_{(p)}\) where G is the special linear group of a finite field, (P) denotes localization and \(+\) denotes Quillen's ''plus construction''. In the second part of the paper the authors show that if a delooping exists, then n divides p-1.