# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.