A quotient of the set [BG,BU(n)] for a finite group G of small rank

Abstract

Let Gp be a Sylow p-subgroup of the ÿnite group G and let Char G n (Gp) represent the set of degree n complex characters of Gp that are the restrictions of class functions on G. We construct a natural map G : [BG; BU (n)] → p G| Char G n (Gp) and prove that G is a surjection for all ÿnite groups G that do not contain a subgroup isomorphic to (Z=p) 3 for any prime p. We show, furthermore, that G is in fact a bijection for two types of ÿnite groups G: those with periodic cohomology and those of odd order that do not contain a subgroup isomorphic to (Z=p) 3 for any prime p.

Vector bundles over classifying spaces of p‐local finite groups and Benson–Carlson duality

“…This description coincides with the one given in Cantarero-Castellana [8,Section 3]. When dealing with a finite group G, this is described in Jackson [18].…”

Cellularization of classifying spaces and fusion properties of finite groups

“…We think that they might be of independent interest, and they also give an alternative proof of the presence of p-torsion in the BZ/p-cellularization of the classifying spaces. The map we are looking for is best understood as a fusion preserving representation of the Sylow p-subgroup of G in some compact Lie group (this relationship is well explained by Jackson in [23]). This observation definitively shifts the problem to the study of fusion systems, a notion which has recently led Broto, Levi, and Oliver to the concept of p-local finite groups in a topological context (see [8,9]).…”

Qd(p)-free rank two finite groups act freely on a homotopy product of two spheres

“…Theorem 15 (Jackson [20,Theorem 1.3]). If G is a finite group that does not contain a rank three elementary abelian subgroup, then the natural mapping…”

“…By hypothesis, Z is not strongly closed in G p with respect to G. Recall that Alperin's Fusion Theory [3] states that given a weak conjugation family F, there exists an (S, T ) ∈ F with Z ⊆ S ⊆ G p and with Z not strongly closed in S with respect to T . The collection of all pairs (P, N G (P)), where P ⊆ G p is a principal p-radical subgroup of G, is the weak conjugation family of Goldschmidt [15,Theorem 3.4] (see also [20]). There exists P ⊂ G p that is a principal p-radical subgroup of G with Z ⊆ P and with Z not strongly closed in P with respect to N G (P).…”