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Projective modules, filtrations and Cartan invariants. (English) Zbl 0533.20003
Let \(G\) be a finite group, \(N\) a normal subgroup of \(G\), \(k\) some field of prime characteristic \(p\). Put \(\overline G=G/N\). Any \(k\overline G\)-module may be viewed as a \(kG\)-module. In particular, any simple \(k\overline G\)-module will have projective covers both as \(k\overline G\)-module and as \(kG\)-module. In this paper the relation between these two modules is investigated. The main result is: Theorem. There is a \(kG\)-module \(M\) such that, whenever \(S\) is a simple \(k\overline G\)-module and \(S\) has projective covers \(Q\) and \(\overline Q\) as a \(kG\)-module and as a \(k\overline G\)-module respectively, then \(Q\) and \(\overline Q\otimes M\) have the same composition factors. As a corollary a new proof is obtained of a theorem of Brauer on Cartan invariants.
Reviewer: R.W.van der Waall

MSC:
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C15 Ordinary representations and characters
20C20 Modular representations and characters
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