On p-radical groups
type: Item
instance of: scholarly article
title: On p-radical groups
zbMATH Open document ID: 0597.16011
DOI: 10.1016/0021-8693(86)90169-9
author: Y. Tsushima
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let G be a finite group and let K be a field of characteristic \(p>0\). Let JK(G) be the Jacobson radical of the group ring K(G). Let P be a p-Sylow subgroup of G. G is said to be p-radical if \(\cap_{x\in G}K(G)JK(P^ x)=JK(G)\). [Cf. \textit{W. Feit}, The Representation Theory of Finite Groups (1982; Zbl 0493.20007), pp. 265-268.] If G is p-radical and H is a normal subgroup of G then H and G/H are p- radical groups. If H is a normal p-subgroup of G and G/H is p-radical then G is p-radical. \textit{T. Okuyama} has shown that a p-radical group is necessarily p-solvable (to appear). The author shows that if G has an abelian p-complement then G is p-radical. The main result, established using block and character theory (not all notation is explained), is that if \(G=PH\) is p-nilpotent where \(H=O_{p'}(G)\) then G is p-radical if and only if \([K,D]\cap C_ K(D)=1\) for any p-subgroup of G.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 16Nxx
zbMATH DE Number: 3961873
zbMATH Keywords: Jacobson radical
zbMATH Keywords: group ring
zbMATH Keywords: p-Sylow subgroup
zbMATH Keywords: p-radical groups
zbMATH Keywords: p- solvable
zbMATH Keywords: p-complement
zbMATH Keywords: p-nilpotent
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90169-9
OpenAlex ID: W2015891718
cites work: Q3955560
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description: scientific article; zbMATH DE number 3961873