A condition on lengths of conjugacy classes and character degrees
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.
scientific article; zbMATH DE number 902862