Linear characters of Sylow subgroups.
type: Item
instance of: scholarly article
title: Linear characters of Sylow subgroups.
zbMATH Open document ID: 1037.20007
DOI: 10.1016/S0021-8693(03)00391-0
author: Gabriel Navarro
published in: Journal of Algebra
publication date: 2003-12-04T00:00:00Z
review text: In this paper, the author was motivated by considering the McKay conjecture for finite groups with a self-normalizing Sylow \(p\)-subgroup. If \(p\geq 5\), the author clarifies the situation. Here are a few of the results in the paper: Theorem A. If \(\lambda,\mu\) are linear characters of a finite \(p\)-group \(P\) with \(p\) an odd prime and \(\lambda=\mu\) up to induced characters of proper subgroups, then \(\lambda=\mu\). Corollary B. Let \(P\) be a Sylow \(p\)-subgroup of \(G\) with \(p\) odd. If \(P\) is self-normalizing, \(\lambda,\mu\) are linear characters of \(P\) and \(\lambda^G=\mu^G\), then \(\lambda=\mu\). The author shows that certain nice class functions are characters (Theorem~D and Corollary~E). Theorem F. Suppose that \(G\) is \(p\)-solvable and \(P\) is a self-normalizing Sylow \(p\)-subgroup. (a) If \(\lambda\) is a linear character of \(P\), then \(\lambda^G\) is the sum of an irreducible character of degree prime to \(p\) and irreducible characters of degree divisible by \(p\). (b) Let \(\chi\) be an irreducible character of \(G\) of degree prime to \(p\). Then \(\chi_P\) is the sum of a linear character and nonlinear irreducible characters of \(P\). Furthermore the map \(\lambda\to\chi\) is a bijection of the linear characters of \(P\) with the set of irreducible characters of \(G\) of degree prime to \(p\). Theorem F is false without assuming solvability (for \(p=2\)). Moreover, by a result of the author, the reviewer and \textit{G. Malle}, Self-normalizing Sylow subgroups [Proc. Am. Math. Soc. 132, No. 4, 973-979 (2004; Zbl 1049.20010)], \(P\) self-normalizing implies that \(G\) is solvable (using the classification of finite simple groups).
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 2013030
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: linear characters
zbMATH Keywords: McKay conjecture
zbMATH Keywords: irreducible characters
zbMATH Keywords: induced characters
reviewed by: Robert Guralnick
MaRDI profile type: Publication
cites work: Characters of solvable groups induced by linear characters of Hall subgroups
cites work: Self-normalizing Sylow subgroups
cites work: Fong characters in \(\pi\)-separable groups
cites work: Q4012102
cites work: Q4312077
cites work: Characters of \(p'\)-degree of \(p\)-solvable groups
cites work: Classical theory of algebraic numbers
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description: scientific article; zbMATH DE number 2013030