The Projective Cover of the Trivial Module Over a Group Algebra of a Finite Group

Abstract

We determine all finite groups G such that the Loewy length (socle length) of the projective cover P(k _G ) of the trivial kG-module k _G is four, where k is a field of characteristic p > 0 and kG is the group algebra of G over k, by using previous results and also the classification of finite simple groups. As a by-product we prove also that if p = 2 then all finite groups G such that the Loewy lengths of the principal block algebras of kG are four, are determined.

Key Words:

• Conjugacy class of involutions
• Loewy length
• Principal block
• Projective cover
• Socle length

2010 Mathematics Subject Classification:

• 20C20
• 20C05

ACKNOWLEDGMENTS

The author is grateful to the referee for reading the first draft carefully and for giving useful comments and remarks. The author would like to thank Professor Tetsuro Okuyama who kindly informed the author Theorem 4.1. Actually the main results in this paper came up to the author via a very useful discussion with Professor Tetsuro Okuyama. The author is grateful to Professors Burkhard Külshammer and Ronald Solomon for pointing out mistakes in the references and the proof of Lemma 5.1, respectively.

Notes

Communicated by A. Tarull.