Endotrivial modules: a reduction to $p'$-central extensions

Caroline Lassueur and Jacques Thévenaz

Vol. 287 (2017), No. 2, 423–438

DOI: 10.2140/pjm.2017.287.423

Abstract

We examine how, in prime characteristic $p$, the group of endotrivial modules of a finite group $G$ and the group of endotrivial modules of a quotient of $G$ modulo a normal subgroup of order prime to $p$ are related. There is always an inflation map, but examples show that this map is in general not surjective. We prove that the situation is controlled by a single central extension, namely, the central extension given by a $p^{\prime }$-representation group of the quotient of $G$ by its largest normal $p^{\prime }$-subgroup.

Keywords

Endotrivial modules, Schur multipliers, central extensions, perfect groups

Mathematical Subject Classification 2010

Primary: 20C20

Secondary: 20C25

Milestones

Received: 4 January 2016

Revised: 26 May 2016

Accepted: 7 September 2016

Published: 9 March 2017