Sharp characters and prime graphs of finite groups
Let \(G\) be a finite group and \(\chi\) a virtual character of \(G\). Then \(| G|\) divides \(\prod_{x \in L^*} (\chi(1) - x)\), where \(L^* := \{\chi(g) \mid g\in G\setminus \{1\}\}\). We say that \(\chi\) is sharp (of type \(L^*\)) if \(| G| = \prod_{x \in L^*} (\chi(1) - x)\). In this paper relations between the structure of \(G\) and its sharp characters are studied, also in connection with graph theory. The main purpose of the paper is to prove the following Theorem 1. Let \(G\) be a finite group. Then equivalent are: (a) \(2 \leq \text{com}(G)\); (b) \(G\) has a 2-connected sharp ordinary character of rank 2; (c) \(G\) has a 2-connected character of rank 2; (d) the augmentation ideal of \(G\) decomposes as a module. There are more results proven, but we cannot mention them here. Finally we mention for the convenience of the reader: Definition. The prime graph \(\Gamma(G)\) of the finite group \(G\) is the following graph: \(V(\Gamma(G)) = \pi(G)\) and \(xy \in E(\Gamma(G))\) if and only if there exists \(g \in G\) such that \(x \cdot y\) divides \(| g|\). We denote by \(\text{com}(G)\) the number of connected components of \(\Gamma(G)\).
scientific article; zbMATH DE number 501075