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The Projective Cover of the Trivial Module Over a Group Algebra of a Finite Group
Authors:Shigeo Koshitani
Institution:1. Department of Mathematics, Graduate School of Science , Chiba University , Chiba , Japan koshitan@math.s.chiba-u.ac.jp
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.
Keywords:Conjugacy class of involutions  Loewy length  Principal block  Projective cover  Socle length
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