Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Projective Modules, Filtrations and Cartan Invariants

Projective Modules, Filtrations and Cartan Invariants PROJECTIVE MODULES, FILTRATIONS AND CARTAN INVARIANTS J. L. ALPERIN, M. J. COLLINS AND D. A. SIBLEY 1. Introduction There are many results relating normal subgroups and representations of finite groups. We shall establish a theorem about projective modules and normal subgroups and then apply it to give a new proof of a theorem of Brauer on Cartan invariants. Let G be a finite group and let k be a field of prime characteristic p. The indecomposable projective /cG-modules have unique simple quotient modules and this establishes a one-to-one correspondence with the simple /cG-modules: the indecomposable projectives are their projective covers. Now let N be a normal subgroup of G, and put G = G/N. Any /cG-module may be viewed as a /cG-module: in particular, any simple /cG-module will have projective covers both as a fcG-module and as a /cG-module. We ask how these are related: with this notation, we can state our main result. THEOREM. There is a kG-module M such that, whenever S is a simple kG-module and S has projective covers Q and Q as a kG-module and as a kG-module respectively, then Q and Q (x) M have the same composition factors. If the modules Q, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Projective Modules, Filtrations and Cartan Invariants

Loading next page...
 
/lp/wiley/projective-modules-filtrations-and-cartan-invariants-VdRcKJR0qw

References (0)

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/16.4.416
Publisher site
See Article on Publisher Site

Abstract

PROJECTIVE MODULES, FILTRATIONS AND CARTAN INVARIANTS J. L. ALPERIN, M. J. COLLINS AND D. A. SIBLEY 1. Introduction There are many results relating normal subgroups and representations of finite groups. We shall establish a theorem about projective modules and normal subgroups and then apply it to give a new proof of a theorem of Brauer on Cartan invariants. Let G be a finite group and let k be a field of prime characteristic p. The indecomposable projective /cG-modules have unique simple quotient modules and this establishes a one-to-one correspondence with the simple /cG-modules: the indecomposable projectives are their projective covers. Now let N be a normal subgroup of G, and put G = G/N. Any /cG-module may be viewed as a /cG-module: in particular, any simple /cG-module will have projective covers both as a fcG-module and as a /cG-module. We ask how these are related: with this notation, we can state our main result. THEOREM. There is a kG-module M such that, whenever S is a simple kG-module and S has projective covers Q and Q as a kG-module and as a kG-module respectively, then Q and Q (x) M have the same composition factors. If the modules Q,

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jul 1, 1984

There are no references for this article.