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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,
Bulletin of the London Mathematical Society – Wiley
Published: Jul 1, 1984
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