Nested GVZ-groups (Q2629552)

The author introduces the following definitions concerning finite groups \(G\). 1) \(G\) is nested if for all \(\chi,\eta\in \text{Irr}(G)\) with \(\chi(1)\leq\eta(1)\) it holds that \(Z(\eta)\leq Z(\chi)\). 2) A non-abelian \(G\) is a GVZ-group if for all \(\chi\in \text{Irr}(G)\) it holds that \(\chi(g)= 0\) for all \(g\in G\setminus Z(\chi)\). 3) A non-abelian \(G\) is a nested GVZ-group if it is nested and a GVZ-group. In this paper, necessary and sufficient conditions are established for nested groups to be nested GVZ-groups, thereby a generalization of Theorem A in the paper of \textit{A. Fernandez-Alcobar} and \textit{A. Moreto} [Trans. Am. Math. Soc. 353, 2171--2192 (2003)] is provided. Also, a family of nested GVZ-groups is presented, all \(p\)-groups of order \(p^{2n+1}\) with exponent equal to \(p\) and nilpotency class \(n+1\), where \(p\) is a prime greater or all equal to \(n+1\). This paper provides a partial answer to two questions (namely \# 24 and \# 30) of \textit{Y. Berkovich} [Groups of prime power order. Vol. 1. Berlin: Walter de Gruyter (2008; Zbl 1168.20001)]: a) Describe the \(p\)-group \(G\) for which the set \(\{Z(\chi)\mid \chi\in\text{Irr}(G)\}\) is a chain with respect to inclusion; b) Study the \(p\)-groups \(G\) such that \(\chi(1)^2= |G/Z(\chi)|\) for all \(\chi\in\text{Irr}(G)\). In all the above considerations, the notation \(Z(\chi)\) stands for the set \(\{x\in G\mid|\chi(x)|= \chi(1)\}\), where \(\chi\) is an irreducible complex character of \(G\).