A condition on lengths of conjugacy classes and character degrees
type: Item
instance of: scholarly article
title: A condition on lengths of conjugacy classes and character degrees
zbMATH Open document ID: 0872.20006
author: Akihide Hanaki
published in: Osaka Journal of Mathematics
publication date: 1997-10-13T00:00:00Z
review text: E. Bannai investigated the following condition on finite groups. Let \(G\) be a finite group, \(\text{Irr}(G)=\{\chi_i\}_{1\leq i\leq k}\) be the set of all irreducible characters of \(G\), and \(\text{Cl}(G)=\{C_i\}_{1\leq i\leq k}\) be the set of all conjugacy classes of \(G\). Condition. By suitable renumbering \(i\), \(\chi_i(1)^2=|C_i|\), for \(i=1,2,\cdots,k\). We call this condition B-condition (``B'' is due to E. Bannai). A few groups satisfying B-condition are known: abelian groups, Suzuki 2-groups and groups isoclinic to them. In any case, they are nilpotent and their derived lengths are at most 2. In this paper, we shall construct a family of groups satisfying B-condition. Our groups are, in a sense, generalizations of Suzuki 2-groups. By our examples, we can say that derived lengths of groups satisfying B-condition are unbounded.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 902862
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: conjugacy classes
zbMATH Keywords: derived lengths
zbMATH Keywords: generalizations of Suzuki 2-groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 902862