The abelian monoid of fusion-stable finite sets is free

2013/02/19 by Reeh, Sune Precht
#19A22 #20D20 #20J15 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1302.4628

Abstract

We show that the abelian monoid of isomorphism classes of G-stable finite S-sets is free for a finite group G with Sylow p-subgroup S; here a finite S-set is called G-stable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable S-sets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat non-obvious) basis, whose elements are in one-to-one correspondence with the G-conjugacy classes of subgroups in S. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a saturated fusion system into its associated ghost ring.

Related

• On the Basis of the Burnside Ring of a Fusion System
• Realizability and tameness of fusion systems
• Burnside Rings of Fusions Systems and their Unit Groups
• Assigning a classifying space to a fusion system up to F-isomorphism
• Prime ideals of the Burnside ring of a saturated fusion system