On Hadamard property of a certain class of finite groups
type: Item
instance of: scholarly article
title: On Hadamard property of a certain class of finite groups
zbMATH Open document ID: 0911.20008
author: Pan Soo Kim
author: Jung R. Cho
author: Noboru Ito
published in: Journal of Algebra
publication date: 1998-12-08T00:00:00Z
review text: A finite group \(G\) of order \(8n\) with a central involution \(e^*\) is called a Hadamard group if \(G\) contains a transversal \(D\) with respect to \(\langle e^*\rangle\) such that \(| D\cap Dr|=2n\) for every element \(r\) of \(G\) outside \(\langle e^*\rangle\). Such a transversal is called a Hadamard subset. The cyclic group of order 4 is considered as a Hadamard group. If \(r^2=e^*\), then we have \(| D\cap Dr|=2n\) for every transversal \(D\) of \(G\) with respect to \(\langle e^*\rangle\). Let \(A(e^*)\) be the set of elements \(r\) of \(G\) such that \(r^2=e^*\) and \(a(e^*)=| A(e^*)|\). Then if \(a(e^*)\) is large enough compared with \(| G|\) we may expect \(G\) to be Hadamard. In the present paper one assumes that \(a(e^*)\geq 4n\) and we investigate the Hadamard property of \(G\) (of order \(8n\)). Let \(T(G)\) be the sum of the irreducible characters of \(G\). Then \(T(G)\leq a(e^*)\), by the Frobenius-Schur formula on the number of involutions. This means that \(2T(G)\geq| G|\), and such groups were classified by the reviewer and K. G. Nekrasov [see \textit{Ya. G. Berkovich} and \textit{E. M. Zhmud}, Characters of finite groups, Part 1. Transl. Math. Monogr. 172, Am. Math. Soc., Providence (1998), Chapter 11]. The authors divide their groups into three classes: Class I consists of groups such that \(a(e^*)>4n\), class II consists of groups \(G\) such that \(a(e^*)=4n\) and \(G\) contains a nonreal element, and class III consists of groups \(G\) such that \(a(e^*)=4n\) and all elements of \(G\) are real. For groups of classes I and II we have that \(2T(G)>| G|\). Note that generalized quaternion groups are members of class I (it is known that these groups are Hadamard; semidihedral and dihedral groups are not Hadamard). The main result of this paper is the following Proposition 3. Any 2-group of order \(8n\) with \(a(e^*)\geq 4n\) is Hadamard. Note that there exists a non-Hadamard group among groups of class II. Namely, class II contains the series o…
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 05B20
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05B10
zbMATH DE Number: 1218679
zbMATH Keywords: finite groups
zbMATH Keywords: central involutions
zbMATH Keywords: transversals
zbMATH Keywords: irreducible characters
zbMATH Keywords: generalized quaternion groups
zbMATH Keywords: Hadamard groups
zbMATH Keywords: Hadamard 2-groups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: jabr.1997.7371
OpenAlex ID: W2009293297
cites work: Q4842717
cites work: Q4359577
cites work: Q5551256
cites work: Cocyclic Hadamard matrices and Hadamard groups are equivalent
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cites work: REMARKS ON HADAMARD GROUPS
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DOI: 10.1006/JABR.1997.7371
description: scientific article; zbMATH DE number 1218679