University of Isfahan International Journal of Group Theory 2251-7650 15 4 2026 12 17 Problems on Brauer characters 179 192 30141 10.22108/ijgt.2025.147244.1995 EN Yanjun Liu School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, China Wolfgang Willems Fakultat fur Mathematik, Otto-von-Guericke Universitat, Magdeburg, Germany and Departamento de Matematicas Universidad del Norte, Barranquilla, Colombia Jiping Zhang School of Mathematical Sciences, Peking University, Beijing, China Journal Article 2025 10 28 In this paper we focus on problems of irreducible $p$-Brauer characters and state conjectures based on many examples which we computed. Many of the asked questions hold true for $p$-solvable groups, but their answers in general seem to require a much deeper understanding than we have at the moment. The questions are dealing with degrees, Hilbert divisors, divisibility, height-zero irreducible Brauer characters, number of irreducible Brauer characters in a $p$-block, and Cartan invariants. For instance, one of the main conjectures states that an irreducible Brauer character has Hilbert divisor 1 if and only if the character lies in a $p$-block of defect zero. This is true for $p$-solvable groups since in this case there is a relation between Hilbert divisors and vertices. However, even for a $p$-block of a non-$p$-solvable group containing only two irreducible Brauer characters we do not have an idea how to attack the problem. We hope that the conjectures and questions which we state in this paper will inspire further research. https://ijgt.ui.ac.ir/article_30141_b4f71d4959352b7f4a42f1421635ff44.pdf