Resumen de Regular orbits of actions of finite soluble groups. Applications

Hangyang Meng

• The main goal of the thesis is to prove some results about the existence of regular orbits of actions of finite soluble groups and apply them to obtain results that can be considered significant steps to solve the Gluck conjecture on large character degrees and questions posed by Kamornikov, Shemetkov and Vasil'ev in the Kourovka Notebook about intersections of system normalisers and prefrattini subgroups of finite soluble groups. We prove that if a finite soluble group G has a faithful and completely irreducible module V and H is a subgroup of G such that VH has is S_4-free, then H has at least two regular orbits on V x V. If H is supersoluble, then H has at least one regular orbit on V x V. This theorem extends all previous results on regular orbits have some significant applications in Finite Group Theory.