On a conjecture concerning character degrees of some \(p\)-groups
type: Item
instance of: scholarly article
title: On a conjecture concerning character degrees of some \(p\)-groups
zbMATH Open document ID: 0860.20013
DOI: 10.1007/BF01198066
author: Andrea Previtali
published in: Archiv der Mathematik
publication date: 1997-02-20T00:00:00Z
review text: It has been conjectured that the character degrees of \(P\in\text{Syl}_p(A_n(q))\), \(q\) a \(p\)-power, are \(q\)-powers. This has recently been established by \textit{I. M. Isaacs} [J. Algebra 177, No. 3, 708-730 (1995; Zbl 0839.20010)]. We extend this result to the \(p\)-Sylow subgroups of \(C_n(q)\), \(D_n(q)\) and \({^2A}_{2n-1}(q^2)\), \(p\) an odd prime. This result was proved by \textit{D. Kazhdan} [Isr. J. Math. 28, 272-286 (1977; Zbl 0391.22006)] under the restriction \(n<p\), which allows him to define the exponential map on the Lie algebra \(L(P)\) associated to \(P\). On the other hand, Isaacs shows that the character degrees of \(P\) are \(q\)-powers, whenever \(P\) is an algebra group over \(\mathbb{F}_q\). Unfortunately, it is not clear whether this applies to our cases. To solve this problem, we define a map sharing some of the properties of the exponential map, with the advantage of being defined whenever \(p\) is odd, and prove that some suitable sections of \(P\) are strong subgroups.
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 937024
zbMATH Keywords: character degrees
zbMATH Keywords: \(p\)-Sylow subgroups
zbMATH Keywords: Lie algebras
zbMATH Keywords: exponential maps
zbMATH Keywords: sections
zbMATH Keywords: strong subgroups
Wikidata QID: Q122928784
MaRDI profile type: Publication
cites work: Characters of groups associated with finite algebras
cites work: Proof of Springer's hypothesis
full work available at URL: https://doi.org/10.1007/bf01198066
OpenAlex ID: W2040127796
description: scientific article; zbMATH DE number 937024