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Some remarks on permutation modules. (English) Zbl 0661.20007

Let k be an algebraically closed field of characteristic \(p>0\), G be a finite group, \(S\in Syl_ p(G)\). In the first section the author shows that the endomorphism ring of the permutation module \(Ind^ G_ S(k)\) holds much information about the fusion of p-subgroups of G as well as information about the p-modular representations of G. The non-trivial subgroups of S which occur as vertices of indecomposable summands of \(Ind^ G_ S(k)\) constitute a conjugate family for S in G. Next he gives a new ring theoretic proof of Frobenius’ normal p-complement theorem. In the second section he studies the simplicial complex \(\Delta_ p\), associated to the partially ordered set of non-trivial p- subgroups of G, and shows that a certain virtual module (naturally associated to \(\Delta_ p)\) is a virtual projective module [see P. J. Webb, J. Algebra 104, 351-357 (1986; Zbl 0606.20042)]. It is proved that if \(\Delta_ p\) is contractible, then every hyperelementary \(p'\)- subgroup of G normalizes some non-trivial p-subgroup of G. A finite group H is called hyperelementary if it is an extension of cyclic group by primary group (Berman-Witt).
Reviewer: Ya.G.Berkovich

MSC:

20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20J05 Homological methods in group theory
20D30 Series and lattices of subgroups

Citations:

Zbl 0606.20042
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References:

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