×

A remark on p-radical groups. (English) Zbl 0713.20004

Let FG be the group algebra of a finite group G over an algebraically closed field F of characteristic \(p>0\). Let P be a p-Sylow subgroup of G. Following K. Motose and Y. Ninomiya [Math. J. Okayama Univ. 17, 171-176 (1975; Zbl 0316.20004)], G is said to be p-radical if the induced module \((F_ P)^ G\) from the trivial FP-module \(F_ P\) is a direct sum of simple FG-modules. T. Okuyama proved that a p-radical group is p-soluble [Hokkaido Math. J. 10, 299-318 (1981; Zbl 0488.20013)]. The author derives a useful sufficient condition that G should be p-radical, namely that for any simple FG-module S whose vertex is Q the restriction to \(F[N_ G(Q)]\) should be a simple \(F[N_ G(Q)]\)- module.
Reviewer: D.A.R.Wallace

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20C20 Modular representations and characters
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alperin, J. L., Weights for finite groups, (The Arcata Conference on Representations of Finite Groups. The Arcata Conference on Representations of Finite Groups, Proceedings of Symposia in Pure Mathematics, Vol. 47 (1987), Amer. Math. Soc.,: Amer. Math. Soc., Providence), 369-379, Part. 1 · Zbl 0657.20013
[2] Feit, W., The Representation Theory of Finite Groups (1982), North-Holland: North-Holland Amsterdam · Zbl 0493.20007
[3] Gorenstein, D., Finite Groups (1980), Chelsea: Chelsea New York · Zbl 0185.05701
[4] Hamernik, W.; Michler, G., On vertices of simple modules in \(p\)-solvable groups, (Mitt. Math. Sem. Giessen, 121 (1976)), 147-162
[5] Knörr, R., On the vertices of irreducible modules, Ann. of Math. (2), 110, 487-499 (1979) · Zbl 0388.20004
[6] Landrock, P., Finite Group Algebras and their Modules, (London Mathematical Society Lecture Note Series, Vol. 84 (1983), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, U.K.) · Zbl 0523.20001
[7] Motose, K.; Ninomiya, Y., On the subgroups \(H\) of a group \(G\) such that \(J\)(KH) KG ⊃ \(J\)(KG), Math. J. Okayama Univ., 17, 171-176 (1975) · Zbl 0316.20004
[8] Okuyama, T., Module correspondence in finite groups, Hokkaido Math. J., 10, 299-318 (1981) · Zbl 0488.20013
[9] Okuyama, T., \(p\)-Radical groups are \(p\)-solvable, Osaka J. Math., 23, 467-469 (1986) · Zbl 0611.20006
[10] Tsushima, Y., On \(p\)-radical groups, J. Algebra, 103, 80-86 (1986) · Zbl 0597.16011
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.