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On the principal indecomposables of a modular group algebra. (English) Zbl 0532.20004
In this paper homological methods are applied to obtain module-theoretic results in the modular representation theory of finite groups. The flavour of the paper is best given by quoting a few of the author’s results. Let G be a finite group and let k be a field of characteristic p. Let J be the Jacobson radical of kG and let P be the principal indecomposable module associated with a simple kG-module A. It is proved that \(O_{p'p}(CA)\), where CA is the centralizer of A, is the intersection of the centralizers of the composition factors of \(P/J^ 2P\), thereby giving a direct proof of H. Pahling’s improvement [Mitt. Math. Semin. Gießen 149, 107-113 (1981; Zbl 0475.20011)] of a well-known result of G. Michler [Proc. Am. Math. Soc. 37, 47-49 (1973; Zbl 0233.16013)] and W. Willems [Math. Z. 171, 163-174 (1980; Zbl 0435.20005)]. Denoting the Frattini subgroup of \(G/O_{p'}G\) by \(\Phi G/O_{p'}G\) it is shown that the centralizer of \(P/J^ 2P\) is contained in \(\Phi\) G. Generalizing a result of W. Gaschütz [Math. Z. 58, 160-170 (1953; Zbl 0050.022)] it is shown that if G is p- solvable and if A, B are both simple kG-modules such that C\(A\neq CB\) then \(Ext(A,B)=Hom_{kG}(K\otimes A,B)\) where \(K=CA\cap CB\); results are also obtained when \(CA=CB\). If, in particular, G has p-length 1 and if A, B are simple kG-modules in the principal block of kG then \(Ext_{kG}(A,B)=Hom_{kG}(C\otimes A,B)\) where \(C=O_{p'p}G\).
Reviewer: D.A.R.Wallace

20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D25 Special subgroups (Frattini, Fitting, etc.)
Full Text: DOI
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