The Influence Of Quantitative Properties Of Conjugacy Subclass Sizes On The Structure Of Factorised Groups

The influence of quantitative properties of conjugacy class sizes on the structure of finite groups is an important subject in finite group theory.Let G be a finite group and x be an element of G,we call the conjugacy class of x in its subnormal closure the conjugacy subclass of x in group G.Obviously,the concept is a generalization of the concept of conjugacy class.In this paper,by using of the concept of conjugacy subclass sizes,we study the influence of the quantitative properties of conjugacy subclass sizes on the structure of the factorised group G=AB（where A and B are subgroups of G）,thus,some conclusions related to conjugacy class sizes are generalized.Precisely,we mainly study the following two aspects:（a）We study the influence on the structure of factorised group G=AB under the condition that conjugacy subclass sizes of some elements is square free.（b）We study the effect of some special elements（such as π-elements）with specific conjugacy subclass sizes（such as π-number or π’-number）on the π-length and the the normality of Hall π-subgroups of factorised group G=AB.For（a）,in chapter 3 and chapter 4,by assuming that the subgroup A is subnormal in G or the subgroups A and B are mutually permutable,we study the influence of the quantitative properties of the conjugacy subclass sizes of some prime power order p’-elements in A ∪ B on the solvability,the p-nilpotence and the supersolvability of G.For（b）,in chapter 5,under the condition that A is subnormal or G is π-solvable,we study the influence of the conjugacy subclass sizes on the structure of factorised group G=AB.Precisely,with A（?）G,we study the effect of the conjugacy subclass sizes of the π-elements in A U B being π’-number on the π-length of G;when G is π-solvable,we study the relationship between the conjugacy subclass sizes of the π-elements in A∪B being π-number and the normality of Hall π-subgroups of G.