The following paper has been published:
El siguiente artículo ha sido publicado:
El següent article ha sigut publicat:

Adolfo Ballester-Bolinches, Ramon Esteban-Romero, and H. Meng.
On large orbits of actions of finite soluble groups: applications.
Recent advances in pure and applied mathematics. Based on contributions presented at the Second Joint Meeting Spain-Brazil in Mathematics, Cádiz, Spain, December 11–14, 2018, pages 105–113. Cham: Springer, 2020.

doi:10.1007/978-3-030-41321-7_8

Abstract

The main aim of this survey paper is to present two orbit theorems and to show how to apply them to obtain a result that can be regarded as a significant step towards the solution of Gluck’s conjecture on large character degrees of finite soluble groups. We also show how to apply them to solve questions about intersections of some conjugacy families of subgroups of finite soluble groups.

2020 Mathematics Subject Classification: 20C15, 20D10, 20D20, 20D45

Keywords: finite groups, soluble groups, linear groups, regular orbits, formations, prefrattini subgroups, system normalisers