The Steinberg component of a finite group with a split (B,N)-pair
type: Item
instance of: scholarly article
title: The Steinberg component of a finite group with a split (B,N)-pair
zbMATH Open document ID: 0633.20028
DOI: 10.1016/0021-8693(86)90242-5
author: Nalsey B. Tinberg
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(G=(G,B,N,R,U)\) be a finite group with an unsaturated split (B,N)- pair of characteristic p (prime) and rank n [see \textit{C. W. Curtis}, Lect. Notes Math. 131, B1-B39 (1970; Zbl 0233.20005) p. B-8]. Here \(B=UH\) is a semidirect product where U is a normal p-subgroup of B and H is abelian with order prime to p. The Weyl group \(W=N/H\) is generated by the set \(R=\{w_ 1,...,w_ n\}\) of involutions. Let \(\cap_{n\in N}n^{- 1}Un\neq \{1\}\). Let \(\nu\) : \(N\to W\) be the natural epimorphism with \(\nu^{-1}(w)=(w)\) for \(w\in W\). Let F be an algebraically closed field of characteristic r (prime or zero). Let \(Y=Ind^ G_ U(F_ U)\) where \(F_ U\) is F treated as a trivial U-module and let \([T]=\sum_{t\in T}\) \(t\in FG\) for any subset T of G. In [Can. J. Math. 9, 347-351 (1957; Zbl 0079.256)] \textit{R. Steinberg} defined the module FGe where \(e=\sum_{w\in W}(-1)^{\ell (w)}(w)[B]\) and proved that FGe is simple when \(r=0\) or r does not divide \(| G:B|\). In this paper the author finds the socle of the Steinberg module in arbitrary characteristic by studying the indecomposable components of Y; in particular, by examining what he calls the Steinberg component.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 4029829
zbMATH Keywords: finite group with an unsaturated split (B,N)-pair of characteristic p
zbMATH Keywords: semidirect product
zbMATH Keywords: Weyl group
zbMATH Keywords: involutions
zbMATH Keywords: Steinberg module
zbMATH Keywords: indecomposable components
zbMATH Keywords: Steinberg component
reviewed by: Andreiy Kondrat'yev
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90242-5
OpenAlex ID: W2015062705
cites work: Q5641300
cites work: On a Theorem of H. Sawada
cites work: A characterization of the modular representations of finite groups with split \((B,N)\)-pairs
cites work: Prime Power Representations of Finite Linear Groups II
cites work: Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs
cites work: Some indecomposable modules of groups with split (B,N)-pairs
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description: scientific article; zbMATH DE number 4029829