Iiyori, Nobuo Sharp characters and prime graphs of finite groups. (English) Zbl 0815.20007 J. Algebra 163, No. 1, 1-8 (1994). 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 \(\Gamma(G)\). Reviewer: R.W.van der Waall (Amsterdam) Cited in 4 Documents MSC: 20C15 Ordinary representations and characters 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20C05 Group rings of finite groups and their modules (group-theoretic aspects) Keywords:finite group; virtual character; sharp characters; sharp ordinary characters; augmentation ideal; prime graph PDFBibTeX XMLCite \textit{N. Iiyori}, J. Algebra 163, No. 1, 1--8 (1994; Zbl 0815.20007) Full Text: DOI