2008/09/02 by Serge Bouc, Bouc, Serge
Computer Science · Mathematics · #20C20 #Advanced Algebra and Logic #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras #math.GR #msc:20C20
paper · pdf · doi:10.48550/arxiv.0809.0493
arxiv created 2008/09/02 · openalex publication_date 2008/09/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, I show that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups 0→ T(P)→ D(P)→\lprojP→ H1(\apdeux(P),\Z)(P), where D(P) is the Dade group of P and T(P) is the subgroup of endo-trivial modules. Here \lprojP is the group of sequences of compatible elements in the Dade groups D(NP(Q)/Q) for non trivial subgroups Q of P. The poset \apdeux(P) is the set of elementary abelian subgroups of rank at least 2 of P, ordered by inclusion. The group H1(\apdeux(P),\Z)(P) is the subgroup of H1(\apdeux(P),\Z) consisting of classes of P-invariant 1-cocycles. Here \lprojP is the group of sequences of compatible elements in the Dade groups D(NP(Q)/Q) for non trivial subgroups Q of P. The poset \apdeux(P) is the set of elementary abelian subgroups of rank at least 2 of P, ordered by inclusion. The group H1(\apdeux(P),\Z)(P) is the subgroup of H1(\apdeux(P),\Z) consisting of classes of P-invariant 1-cocycles. A key result to prove that the above sequence is exact is a characterization of elements of 2D(P) by sequences of integers, indexed by sections (T,S) of P such that T/S≅ (\Z/p\Z)2, fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank~3, or extraspecial of order p3 and exponent p.