On Camina groups of prime power order
type: Item
instance of: scholarly article
title: On Camina groups of prime power order
zbMATH Open document ID: 0860.20017
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: The authors define a `Camina group' as one in which \(|C_G(g)|=|C_{G/G'}(gG')|\) for all \(g\) in \(G\setminus G'\). The main result is that if \(G\) is a finite Camina group then \(G\) has nilpotency class at most 3. The proof of the result is contained in Section 3 which is a very neat argument using earlier work of \textit{L. Verardi} [Ann. Mat. Pura Appl., IV. Ser. 148, 131-171 (1987; Zbl 0648.20032)] and \textit{A. Mann} and \textit{C. M. Scoppola} [Arch. Math. 56, No. 4, 320-332 (1991; Zbl 0788.20010)] to get bounds on the orders of various factors of \(G\). They then use a Lie algebra argument to complete the proof. Section 4 is devoted to the construction of class 3 Camina groups. In this they return to the idea in Section 3 to construct their examples and show how the existence of certain skew-symmetric matrices, where entries are homogeneous polynomials, give rise to the examples they are looking for.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 883786
zbMATH Keywords: finite Camina groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: class 3 Camina groups
author: Rex Dark
author: Carlo Maria Scoppola
reviewed by: A. R. Camina
MaRDI profile type: Publication
full work available at URL: jabr.1996.0146
OpenAlex ID: W1972778573
DOI: 10.1006/JABR.1996.0146
Recommended article: Q4544915
Recommended article: A Contribution to the Theory of Groups of Prime-Power Order
Recommended article: Q5259373
Recommended article: On the orders of primitive groups
Recommended article: The Fixity of Groups of Prime-Power Order
Recommended article: Groups of prime power order with many conjugacy classes
Recommended article: A cohomological approach to theory of groups of prime power order
description: scientific article; zbMATH DE number 883786