On \(M\)-groups with Sylow towers.
type: Item
instance of: scholarly article
title: On \(M\)-groups with Sylow towers.
zbMATH Open document ID: 1336.20008
author: Xuewu Chang
author: Huijuan Zheng
author: Ping Jin
published in: Archiv der Mathematik
publication date: 2015-12-11T00:00:00Z
review text: A finite group is called an \(M\)-group, whenever each irreducible character of representations over \(\mathbb C\) is induced by a linear character of a subgroup of that group. This paper handles some theorems, centering around two problems, namely 1) Is a normal subgroup of an \(M\)-group itself an \(M\)-group? 2) Is a Hall-subgroup of an \(M\)-group itself an \(M\)-group? It is well-known, that the answer to both these questions is: not always. As to 1), view \textit{E. C. Dade} [Math. Z. 133, 313-317 (1973; Zbl 0255.20007)] and the reviewer [Nederl. Akad. Wet., Proc., Ser. A 77, 157-167 (1974; Zbl 0279.20013)]; as to 2) view \textit{H. Fukushima} [Proc. Am. Math. Soc. 133, No. 3, 671-675 (2005; Zbl 1088.20003)]. The authors provide in this paper special situations for an \(M\)-group for which that group has its normal subgroups satisfying the \(M\)-group property or for which that group has its Hall-subgroups satisfying the \(M\)-group property. In order to do so, the authors use Isaacs' theory of character triples and the concepts of the theory of linear limits of \textit{E. Dade} and \textit{M. Loukaki} as provided in [\url{arXiv:math/0412385} (2004)]. Some \(M\)-groups admitting a Sylow tower do satisfy one of the properties just mentioned.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6520099
zbMATH Keywords: finite \(M\)-groups
zbMATH Keywords: monomial characters
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: Sylow towers
zbMATH Keywords: Hall subgroups
zbMATH Keywords: Isaacs character triples
zbMATH Keywords: Dade-Loukaki linear limits
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: s00013-015-0833-7
OpenAlex ID: W1961608490
cites work: Normal subgroups of M-groups need not be M-groups
cites work: Groups all of whose subgroups are M-groups
cites work: M-groups and 2-groups
cites work: Hall subgroups of $M$-groups need not be $M$-groups
cites work: M-Groups with Sylow Towers
cites work: Q4103728
cites work: Characters of subnormal subgroups of M-groups
cites work: M-GROUPS OF ORDER paqb: A THEOREM OF LOUKAKI
cites work: Extendible characters and monomial groups of odd order.
cites work: Character ramification and M-groups
cites work: Q4405074
DOI: 10.1007/S00013-015-0833-7
description: scientific article; zbMATH DE number 6520099