Applications of the theory of Camina groups
type: Item
instance of: scholarly article
title: Applications of the theory of Camina groups
zbMATH Open document ID: 0919.20004
DOI: 10.1142/S0252959999000060
author: Yongcai Ren
published in: Chinese Annals of Mathematics. Series B
publication date: 1999-09-08T00:00:00Z
review text: Let \(C_1=\{1\}\), \(C_2,\dots,C_m,C_{m+1},\dots,C_n\) be all conjugacy classes of a finite group \(G\), where \(2\leq m\leq n-1\). The following hypotheses are considered. Hypothesis (H1). \(G\) has a character \(\chi\) such that (a) \(\chi(C_2)=\cdots=\chi(C_m)\), (b) for every \(\varphi\in\text{Irr}(\chi)\), \(\varphi(C_{m+1})=\cdots=\varphi(C_n)=0\). Hypothesis (H2). \(C_iC_j=C_j\) for \(i=1,\dots,m\) and \(j=m+1,\dots,n\). Theorem 1.1. Hypotheses (H1) and (H2) are equivalent. Let \(G\) satisfy (H1) and (H2). Then \(N=\bigcup^m_{i=1}C_i\) is a normal subgroup of \(G\), the kernel of \(G\). Such groups \(G\) coincide with groups that were studied by \textit{A. R. Camina} [see Isr. J. Math. 31, No. 2, 153-160 (1978; Zbl 0654.20019)]. It follows from Proposition 25.9 and the Remark 1 following it [in \textit{Ya. Berkovich} and \textit{E. Zhmud}, Characters of Finite Groups, Part 2, Transl. Math. Monogr. 181, Am. Math. Soc., Providence (1999)], that the subgroup \(N\) is solvable. In \S 3, a subclass of the class of \(V^3\)-groups, i.e., non-Abelian groups all of whose nonlinear irreducible characters assume exactly three values [see \textit{Ya. Berkovich}, \textit{D. Chillag} and \textit{E. Zhmud}, Houston J. Math. 21, No. 1, 17-28 (1995; Zbl 0833.20008)], is studied. A group \(G\) is said to be a restricted \(V^3\)-group if it is a \(V^3\)-group such that \(\ker(\chi)<G'\) for every nonlinear \(\chi\in\text{Irr}(G)\). In \S 4 the main theorem in [\textit{Ya. Berkovich}, Publ. Math. 44, No. 3-4, 225-234 (1994; Zbl 0840.20004)] is generalized slightly.
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1271837
zbMATH Keywords: conjugacy classes
zbMATH Keywords: finite groups
zbMATH Keywords: characters
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s0252959999000060
OpenAlex ID: W2408531707
Recommended article: Q5259373
Recommended article: Q3375127
Recommended article: Generalizing Camina groups and their character tables
Recommended article: Q3747265
Recommended article: Q3775689
Recommended article: A Frobenius group analog for Camina triples
Recommended article: Classifying Camina groups: a theorem of Dark and Scoppola.
Recommended article: On Infinite Camina Groups
Recommended article: On a covering group theorem and its applications
Recommended article: On Camina groups of prime power order
description: scientific article; zbMATH DE number 1271837