The Roquette category of finite p-groups

2011/11/15 by Serge Bouc, Bouc, Serge
Mathematics · #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #math.CT #math.GR #math.RT

paper · pdf · doi:10.48550/arxiv.1111.3469

arxiv created 2011/11/15 · openalex publication_date 2011/11/15 · arxiv updated 2011/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct summand, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from R_p to abelian groups are exactly the rational p-biset functors. This yields in particular very efficient ways of computing such functors on arbitrary p-groups : this applies to the representation functors R_K, where K is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.

Related

• Fast decomposition of p-groups in the Roquette category, for p>2
• Rational p-biset functors
• The functor of units of Burnside rings for p-groups
• Idempotents of double Burnside algebras, L-enriched bisets, and decomposition of p-biset functors
• The classification of endotrivial complexes