Finite groups with minimal 1-PIM.
type: Item
instance of: scholarly article
title: Finite groups with minimal 1-PIM.
zbMATH Open document ID: 1173.20006
author: Gunter Malle
author: Th. Weigel
published in: Manuscripta Mathematica
publication date: 2008-10-16T00:00:00Z
review text: Let \(G\) be a finite group, let \(\ell\) be a prime number and let \(\mathbb{F}\) be a field of characteristic \(\ell\). We denote by \(\Phi^G_1\) the projective cover of the trivial (left) \(\mathbb{F}[G]\)-module \(\mathbb{F}\), that is the unique (up to isomorphism) projective indecomposable \(\mathbb{F}[G]\)-module with socle (and head) isomorphic to the trivial \(\mathbb{F}[G]\)-module \(\mathbb{F}\). This module is also called the 1-PIM of \(\mathbb{F}[G]\). Obviously, \(\Phi^G_1(\mathbb{F})=\Phi^G_1(\mathbb{F}_\ell)\otimes_{\mathbb{F}_\ell}\mathbb{F}\). It is well-known that the \(\mathbb{F}\)-dimension of \(\Phi_1\) is divisible by \(|G|_\ell\), the \(\ell\)-part of \(|G|\). Define \(c_\ell(G):=\dim_\mathbb{F}(\Phi^G_1)/|G|_\ell\). The main purpose of paper under review is to study pairs \((G,\ell)\) for which \(c_\ell(G)=1\), i.e., finite groups \(G\) and prime numbers \(\ell\) for which \(\Phi^G_1\) has minimal \(\mathbb{F}\)-dimension. In particular, the authors classify the non-Abelian finite simple groups \(G\) and primes \(\ell\) satisfying this identity. As a consequence they show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\). Another consequence is the fact that the validity of this identity for a finite group \(G\) and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell'\)-Hall subgroup for \(G\).
reviewed by: A. Iranmanesh
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5351813
zbMATH Keywords: finite simple groups
zbMATH Keywords: finite soluble groups
zbMATH Keywords: Hall subgroups
zbMATH Keywords: projective covers
MaRDI profile type: Publication
full work available at URL: s00229-008-0176-9
OpenAlex ID: W2030651283
cites work: Q3800275
cites work: Generic Cartan invariants for Chevalley groups
cites work: On the projective characters in characteristic 2 of the groups \(Suz(2^m)\) and \(Sp_4(2^n)\)
cites work: Brauer trees in classical groups
cites work: Projective modules for SL(2,q)
cites work: Projective modules for Sp(4,p) in characteristic p
cites work: Q5715118
cites work: Q4839675
cites work: SMALLEST DEGREES OF REPRESENTATIONS OF EXCEPTIONAL GROUPS OF LIE TYPE
cites work: Height 0 characters of finite groups of Lie type
cites work: Minimal characters of the finite classical groups
cites work: On Ree's Series of Simple Groups
DOI: 10.1007/S00229-008-0176-9
Recommended article: Finite Groups with Π-Supplemented Minimal Subgroups
Recommended article: Minimal nonmodular finite p-groups
Recommended article: Finite \(p\)-groups with few minimal nonabelian subgroups.
Recommended article: Finite minimal non-\(\text{T}_1\)-groups.
Recommended article: Finite groups with few vanishing elements
Recommended article: Q3513473
Recommended article: Finite \(p\)-groups with many minimal nonabelian subgroups.
Recommended article: ON MINIMAL SUBGROUPS OF FINITE GROUPS
Recommended article: Q3437624
Recommended article: Finite groups with complemented minimal p-subgroups
description: scientific article; zbMATH DE number 5351813