Gluing endo-permutation modules

TL;DR

This paper establishes an exact sequence relating the Dade group, endo-trivial modules, and certain cohomology groups for odd prime p and finite p-groups, providing new structural insights.

Contribution

It introduces a new exact sequence connecting Dade groups, endo-trivial modules, and cohomology, with a novel characterization of elements in 2D(P).

Findings

Established an exact sequence involving D(P), T(P), and cohomology groups.

Characterized elements of 2D(P) via integer sequences satisfying specific conditions.

Provided structural insights into endo-permutation modules for odd prime p.

Abstract

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 $$0\to T(P)\to D(P)\to\lproj{P}\to H^1\big(\apdeux(P),\Z\big)^{(P)},$$ where $D(P)$ is the Dade group of $P$ and $T(P)$ is the subgroup of endo-trivial modules. Here $\lproj{P}$ is the group of sequences of compatible elements in the Dade groups $D\big(N_P(Q)/Q\big)$ 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 $H^1\big(\apdeux(P),\Z\big)^{(P)}$ is the subgroup of $H^1\big(\apdeux(P),\Z\big)$ consisting of classes of $P$-invariant 1-cocycles. Here $\lproj{P}$ is the group of sequences of compatible elements in the Dade groups $D\big(N_P(Q)/Q\big)$ 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 $H^1\big(\apdeux(P),\Z\big)^{(P)}$ is the subgroup of $H^1\big(\apdeux(P),\Z\big)$ 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 $T/S\cong (\Z/p\Z)^2$, fulfilling certain conditions associated to subquotients of $P$ which are either elementary abelian of rank~3, or extraspecial of order $p^3$ and exponent $p$.