Some new developments and open questions in the character theory of finite groups

Abstract

The following article consists of two mainly distinct parts. The first three sections deal with module actions on finite vector spaces. We start considering the critical case of quasi-primitive modules and indicate how those techniques apply to obtain bounds for the order of solvable linear groups and regular orbits of solvable primitive permutation groups on the power set. Finally, module actions with large centralizers are investigated. Sections 4–6 are concerned with several central problems in character theory, being Brauer’s height-0 conjecture, the modular Ito problem, the shape of the character degree graph and Huppert’s ρ-σ-conjecture. If the group in question is solvable, then the methods of sections 1–3 can be used to obtain at least partial results. Relying on the classification of finite simple groups, some of those can be extended to p-solvable or even arbitrary finite groups.KeywordsSimple GroupPermutation GroupPrime DivisorSolvable GroupMinimal Normal SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.