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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

MSC:
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.)
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[1] Barnes, D.W.; Schmid, P.; Stammbach, U., Cohomological characterisations of saturated formations and homomorphs of finite groups, Comment. math. helvetici, 53, 165-173, (1978) · Zbl 0383.20032
[2] Curtis, C.W.; Reiner, I., Representation theory of finite groups and associative algebras, (1962), Interscience New York · Zbl 0131.25601
[3] Feit, W., The representation theory of finite groups, (1982), North-Holland Amsterdam · Zbl 0493.20007
[4] Gabriel, P., Indecomposable representations II, Symposia math. inst. naz. alta. mat., 11, 81-104, (1973)
[5] Gaschütz, W., Ueber die ∅-untergruppe endlicher gruppen, Math. Z., 58, 160-170, (1953) · Zbl 0050.02202
[6] Gruenberg, K.W., Groups of non-zero presentation rank, Symp. math., 17, 215-224, (1976) · Zbl 0358.20045
[7] Griess, R.L.; Schmid, P., The Frattini module, Arch. math., 30, 256-266, (1978) · Zbl 0362.20006
[8] Hilton, P.; Stammbach, U., A course in homological algebra, (1970), Springer-Verlag Berlin · Zbl 0238.18006
[9] Huppert, B., Endliche gruppen, I, (1967), Springer-Verlag Berlin · Zbl 0217.07201
[10] Huppert, B.; Blackburn, N., Finite groups, II, (1982), Springer-Verlag Berlin · Zbl 0477.20001
[11] Isaacs, I.; Smith, S., A note on groups of o-length 1, J. algebra, 38, 531-535, (1976) · Zbl 0334.20007
[12] Michler, G., The kernel of a block of a group algebra, Proc. amer. math. soc., 37, 47-49, (1973) · Zbl 0233.16013
[13] Pahlings, H., Normal p-complements and irreducible characters, Math. Z., 154, 243-246, (1977) · Zbl 0337.20009
[14] Pahlings, H., Kerne und projektive auflösungen, Mitt. math. sem. giessen, 149, 107-113, (1981) · Zbl 0475.20011
[15] Stammbach, U., Cohomological characterisations of finite solvable and nilpotent groups, J. pure appl. algebra, 11, 293-301, (1977) · Zbl 0374.20060
[16] Willems, W., On the projectives of a group algebra, Math. Z., 171, 163-174, (1980) · Zbl 0435.20005
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