A sectional characterization of the Dade group

TL;DR

This paper refines the understanding of the Dade group by characterizing it as a limit over specific sections of a finite p-group, extending previous detection results to include more complex sections.

Contribution

It provides a new characterization of the Dade group as a limit over sections that are either elementary abelian of rank ≤3 or extraspecial of order p^3, improving prior detection results.

Findings

Dade group characterized as a limit over specific sections

Includes sections that are elementary abelian of rank ≤3

Incorporates extraspecial sections of order p^3

Abstract

Let $k$ be a field of characteristic $p$, let $P$ be a finite $p$- group, where $p$ is an odd prime, and let $D(P)$ be the Dade group of endo-permutation $kP$-modules. It is known that $D(P)$ is detected via deflation--restriction by the family of all sections of $P$ which are elementary abelian of rank $\leq2$. In this paper, we improve this result by characterizing $D(P)$ as the limit (with respect to deflation--restriction maps and conjugation maps) of all groups $D(T/S)$ where $T/S$ runs through all sections of $P$ which are either elementary abelian of rank $\leq3$ or extraspecial of order $p^3$ and exponent $p$.