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SOME REMARKS ON FILTRATIONS FOR PROJECTIVE MODULES YUN FAN AND BO RONG ZHOU Introduction Let G be a finite group, N^G,G:= G/N, and let F be a field. Alperin, Collins and Sibley [1] obtained a connection between an FG-projective cover P of a simple FG-module V, regarded as an FG-module, with an FG-projective t +l t + cover P of V regarded as an FG-module: PJ /Pf ^P® J /Af \ where J:=J(FN) is the Jacobson radical of FN, A is the augmentation ideal of FN and f/AJ* is a G-conjugation module. Later, Carlson and Collins [2] dropped the hypothesis 'V is simple' and extended the result. In [6], Reynolds started from a classical Clifford's theorem and considered principal indecomposable characters. A special case of the result of [6] is explained as a character version of the module- theoretic result of [1]. It is natural to ask whether the whole result of [6] could be so explained. Recently, Harris [4] has given a nice extension to modules: if the restriction to FN of an FG-module U is absolutely simple and Q is an FJV-projective cover of 1 1 Ui , then Qf/QJ * , J'= J(FN), can be correspondingly
Bulletin of the London Mathematical Society – Wiley
Published: Sep 1, 1992
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