Blocks of small defect in alternating groups and squares of Brauer character degrees
type: Item
instance of: scholarly article
title: Blocks of small defect in alternating groups and squares of Brauer character degrees
zbMATH Open document ID: 1391.20007
author: Xiaoyou Chen
author: James P. Cossey
author: Mark L. Lewis
author: H. P. Tong-Viet
published in: Journal of Group Theory
publication date: 2017-11-06T00:00:00Z
review text: The nilpotency of a finite group was characterized by means of a character theoretic condition by \textit{S. M. Gagola jun.} and \textit{M. L. Lewis} [Commun. Algebra 27, No. 3, 1053--1056 (1999; Zbl 0929.20010)]. They established that a finite group \(G\) is nilpotent if and only if \(\chi(1)^2\) divides \(|G : \text{ker}(\chi)|\) for all \(\chi\in \text{Irr}(G)\). In the paper under review, the authors study a similar divisibility condition for Brauer characters. The main result, Theorem 1, asserts that if \(p\) is a prime and \(G\) be a finite group, then \(\varphi(1)^2\) divides \(|G : \text{ker}(\varphi)|\) for all \(\varphi \in\text{IBr}(G)\) if and only if \(G\) has a normal Sylow \(p\)-subgroup \(P\) and \(G/P\) is nilpotent. The greatest difficulty of the proof of Theorem 1 is to prove that if the divisibility condition holds then the Sylow \(p\)-subgroup is normal in \(G\). The \(p\)-solvable and non-\(p\)-solvable cases are considered separately. For the \(p\)-solvable case, the authors make use, among other techniques, of the B\(_\pi\)-character theory introduced by \textit{I. M. Isaacs} [J. Algebra 86, 98--128 (1984; Zbl 0526.20006)]. In the non-\(p\)-solvable case, the authors manage to reduce the question to whether a certain non-abelian composition factor \(S\) of \(G\) has a defect zero character or not. The former case quickly leads to a contradiction, whereas the second case yields to \(p = 2\) or \(3\) and \(S\) is isomorphic either to certain sporadic simple groups or an alternating group Alt(\(n\)) for \( n \geq 7\). The sporadic groups are handled using GAP and, for the alternating groups, the authors obtain first the following result. Theorem 2. Let \(n\geq 5\) be an integer, let \(p = 2\) or \(3\) and let \(p^a = |\text{Alt}(n)|_p\). Then, \(\text{Alt}(n)\) has a \(p\)-block \(B\) of defect \(d = d(B)\) such that \(d \leq (a - 1)/2\) for \( p = 3\), and \(d = (a - 2)/2\) for \(p = 2\) with the following exceptions: \( n = 7\) if \(p = 3\); a…
reviewed by: Antonio Beltrán
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6803033
zbMATH Keywords: Brauer characters
zbMATH Keywords: blocks
zbMATH Keywords: alternating groups
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: jgth-2017-0025
OpenAlex ID: W2737901363
cites work: Spin Modules for Symmetric Groups
cites work: On \(p\)-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree.
cites work: Q5785002
cites work: Q3684278
cites work: Large orbits in coprime actions of solvable groups
cites work: Blocks of Small Defect
cites work: A Character Theoretic Condition for F(G) > 1
cites work: A character theoretic condition characterizing nilpotent groups
cites work: Defect zero blocks for finite simple groups
cites work: Every coprime linear group admits a base of size two
cites work: Characters of \(\pi\)-separable groups
cites work: Q3808292
cites work: Q4312077
cites work: Large orbits in actions of nilpotent groups
cites work: Q3952291
cites work: Orbit sizes, character degrees and Sylow subgroups.
cites work: Q4387775
cites work: Lower defect groups in symmetric groups
cites work: On the \(p\)-blocks of symmetric and alternating groups and their covering groups
cites work: Q5785003
cites work: Regular orbits of solvable linear \(p'\)-groups.
cites work: Large orbits of supersolvable linear groups
DOI: 10.1515/JGTH-2017-0025
description: scientific article; zbMATH DE number 6803033