A remark on the Jacobson radical of a block ideal in a finite p-solvable group
type: Item
instance of: scholarly article
title: A remark on the Jacobson radical of a block ideal in a finite p-solvable group
zbMATH Open document ID: 0596.16009
DOI: 10.1016/0021-8693(85)90205-4
author: Shigeo Koshitani
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let F be a field of characteristic \(p>0\), G a finite p-solvable group, P a Sylow p-subgroup of G, \(p^ r\) (r\(\geq 2)\) the order of P, FG the group algebra of G over F and J(FG) the Jacobson radical of FG. Let t(G) be the index of nilpotency of J(FG). The author extends his own work for \(p\geq 5\) [J. Algebra 80, 134-144 (1983; Zbl 0507.16011)] and here proves that the following are equivalent: (1) \(t(G)=p^{r-1}+p-1\), (2) \(p^{r- 1}<t(G)<p^ r\), (3) P is noncyclic but has a cyclic subgroup of index p, (4) G has p-length 1 and P is isomorphic either to an Abelian group of type \((p^{r-1},p)\) or to a group of the form \(<a,b|\) \(a^ p=1\), \(b^{p^{r-1}}=1\), \(a^{-1}ba=b^ m>\). The proof depends on showing, by a complicated group-theoretic induction argument, that if \(t(G)>p^{r-1}\) then G has an element of order \(p^{r-1}\). It is observed that for \(p=2\) the main result does not hold.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16Nxx
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 3959687
zbMATH Keywords: finite p-solvable group
zbMATH Keywords: Sylow p-subgroup
zbMATH Keywords: group algebra
zbMATH Keywords: Jacobson radical
zbMATH Keywords: index of nilpotency
zbMATH Keywords: p-length
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90205-4
OpenAlex ID: W1991661707
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description: scientific article; zbMATH DE number 3959687