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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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