Coprime Actions Of Finite Groups And Some Applications

The theory of group actions paly an important role on finite group theory.Many researchers are working in this area and have got lots of nice results.This aim of this thesis is to investigate the theory of coprime actions in finite groups and some applications in abstract group theory and mathematics physics.Most of basic results about group actions are shown in Chapter I.In Chapter II,we study the action which satisfies fixing p-subgroups of some given order.This is an extension of the recent result in[Y.Berkovich and I.M.Isaacs,J.Algebra,2014].As an application,we also give a result about subgroup-embedded theorem.The generalized semiregular actions are investigated in In Chapter III.It is proved that an elementary abelian r-group A acts coprimely on a group G,if C_G?a?is supersoluble for all 1=a?A,then G is supersoluble when|A|?r⁴and G?F₃?G?when|A|?r³.Moreover,we prove some other results for cases when the fixed-point group C_G?a?is abelian,p-nilpotent or satisfies the Sylow tower property.These extend the classical theorem in[J.N.Ward,Bull.Aust.Math.Soc,1971]and the partial results in[P.Shumyatsky,Proc.Amer.Math.Soc,2001].In the last chapter,it is proved by using the theory of coprime actions and nilpotent actions that a finite left brace is left nilpotent if and only if its multi-plicative group is nilpotent,moreover,if the multiplicative group of a finite left brace satisfies the Sylow tower property and is an A-group,then this left brace is right nilpotent.This gives a nice proof of the main result in[A.Smoktunowicz,Tran.Amer.Math.Soc,2018]and extends a theorem in[F.Ced?o,E.Jesper,J.Okninski,Commun.Math.Phys.,2014],also show the connection between the permutation groups and multipermutation solutions of Yang-Baxter equation.