Characters and Sylow 2-subgroups of maximal class revisited
type: Item
instance of: scholarly article
title: Characters and Sylow 2-subgroups of maximal class revisited
zbMATH Open document ID: 1395.20004
author: Gabriel Navarro
author: Benjamin Sambale
author: Pham Huu Tiep
published in: Journal of Pure and Applied Algebra
publication date: 2018-06-06T00:00:00Z
review text: The representation theory of finite groups having Sylow \(2\)-subgroups of maximal class is a classical and widely studied topic. These 2-groups, which are exactly dihedral, semidihedral and generalized quaternion 2-groups, have been characterized from distinct points of views, and particularly, they coincide with the non-abelian 2-groups whose derived subgroup has index 4. In this paper, the authors provide two interesting ways to recognize from the character table whether a finite group \(G\) possesses a Sylow \(2\)-subgroup \(P\) of maximal class. Precisely, the main result establishes the equivalence among the following assertions: (a) \(|P/P'|=4\); (b) either \(|P|=4\) or there exists \(g\in P\) such that \(|G:{\mathbf C}_G(g)|\) is even and \({\mathbb Q}(\chi(g): \chi \in \text{Irr}(G))={\mathbb Q}(\zeta \pm \zeta^{-1})\), where \(\zeta\in {\mathbb C}\) is a primitive \(|P|/2\)-th root of unity; (c) the principal 2-block of \(G\) has exactly four irreducible complex characters of odd degree. The equivalence between properties (a) and (b) is quite elementary. However, the fact that (c) implies (a), which is a particular case of the Alperin-McKay conjecture (unsolved even for \(p=2\)), requires the classification of the finite simple groups. The authors also obtain a highly non-trivial characterization of the finite groups whose Sylow 3-subgroups have order 3. These are those groups whose principal 3-block possesses exactly three irreducible complex characters of degree not divisible by 3. The authors remark, however, that they have not been able to find a characterization of the fact that \(|P/P'|=p^2\) for primes \(p\geq 5\).
reviewed by: Antonio Beltrán
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6881294
zbMATH Keywords: characters
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: character tables
zbMATH Keywords: 2-groups of maximal class
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jpaa.2018.02.002
OpenAlex ID: W2790910168
Wikidata QID: Q58940139
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DOI: 10.1016/J.JPAA.2018.02.002
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description: scientific article; zbMATH DE number 6881294