Real Brauer characters (Q2040520)

Let \(G\) be a finite group and \(p\) a prime dividing the order of \(G\). An element \(x \in G\) is called real if \(x\) is \(G\)-conjugate to its inverse \(x^{-1}\) and an element \(g \in G\) is called \(p\)-regular if \(p\) does not divide the order of \(g\). By Brauer's lemma on character tables, Theorem 6.32 of [\textit{I. M. Isaacs}, Character theory of finite groups. New York-San Francisco-London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers (1976; Zbl 0337.20005)], the number of real-valued irreducible \(p\)-Brauer characters of \(G\) is equal to the number of conjugacy classes of \(p\)-regular real elements of \(G\). The finite groups with exactly two real-valued irreducible \(p\)-Brauer characters are considered in [\textit{G. Navarro} et al., J. Alg. 307 No 2, 891--898 (2007; Zbl 1124.20006)]. The aim of this article is to determine the structure of finite groups with exactly three real-valued irreducible \(p\)-Brauer characters which is equivalent to say that finite groups with exactly three \(p\)-regular real elements. In this article under review, the author first classifies finite simple groups with small number of primes dividing the number of real elements. This is done by attaching a graph to \(G\) whose vertex set is equal to the set of primes dividing the order of real elements of \(G\). Then using this classification, the main results of the paper, namely Theorem A and B, which describe the finite groups with exactly three real-valued irreducible \(p\)-Brauer characters are proved.