On the Average Degree of Characters with Odd or Even Degrees

Abstract

Let acd(2 ' )(C) be the average degree of irreducible characters of C with odd degree. It has been proved that if acd(2 ')(C) < acd(2 ') (A(5)) = 3, then C is a solvable group. On the other hand, let acd(2)(C) be the average degree of linear characters and irreducible characters of C with even degree. It has been shown that if acd(2 ')(C) < acd(2 ')(A(5)) = 5/2, then C is a solvable group. In this paper, we improve these bounds and we show that if C is a finite group with acd(2 ') (C) < acd(2 ') (PSL(2, 7)) = 7/2, then either C is a solvable group or C has a chief factor isomorphic to A5. Also, we prove that if C is a finite group with acd(2)(C) < acd(2 ')(PSL(2, 8)) = 9/2, then either C is a solvable group or all minimal normal subgroups of C are abelian or isomorphic to A5. Clearly, these bounds are the best. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.