Primitive Characters Of Finite Solvable Groups

In this thesis,we study the primitive characters of finite solvable groups,focusing on the existence and uniqueness of multiplicative factorizations of primitive characters,and the associated symplectic modules structure.The goal is to extend some classical theorems about primitive characters to arbitrary irreducible characters,and we expect to establish a multiplicative factorization theorem for a large class of irreducible characters,which can develop more powerful proof techniques to improve or solve several related character problems.As a generalization of primitive characters in solvable groups,we present the concept of C-characters,and describe the absolutely inseparable C-characters,that is,the so-called C_*-characters,which contain Brauer's strong irreducible characters.Also,the Fitting characters and the related uniqueness of Fitting factorizations are defined,and we introduce the symplectic modules and symplectic structures associated with primitive characters.As an application,we obtain the distribution of zeros and value information for C-characters,and the permutation formula of such characters in this thesis,which all generalize the corresponding theorems of Isaacs,Navarro,Ferguson,Turull and Wilde on primitive characters.Specifically,this thesis discusses five interrelated problems of primitive characters.(1)Permutation formulas of primitive characters.With the help of the Isaacs' theory and technique of character fives,a complete characterization of Wilde's permutation formulas for primitive characters is given,and we obtain more structural information of associated subgroups,especially the result that the complement consisting of good elements in the character five associated with a given primitive character is uniqueness up to conjugacy.This is a technical theorem and can be used in many aspects.(2)The distribution of zeros and value information for primitive characters.We investigate the ‘good element' of the character five and obtain a new criterion about it.As an application,we establish three basic properties of C-characters,which generalize the related theorems of Navarro and Wilde on primitive characters,that is,the zeros distribution theorem and the permutation formula.(3)The Fitting factorizations of primitive characters.Given an arbitrary primitive character,it is proved that there always exists the Fitting factorization for this character on some covering group,and such factorization possesses certain uniqueness.(4)The symplectic structure of primitive characters.There exists a correspondence between the multiplicative factorizations of primitive characters and the orthogonal decompositions of their associated symplectic modules,and we obtain the explicitly upper bound of the number of irreducible character factors in multiplicative factorization of the primitive character by means of the associated symplectic structure.In addition,the necessary and sufficient condition for reaching the upper bound is obtained,and a sufficient condition is given for the product of some primitive characters still to be the primitive character.(5)The factorization theorem of primitive characters and its generalizations.In this thesis,we give an effective criterion for C_*-character,and prove that C-character can be factored into the product of several C_*-characters on some covering group.In fact,how to construct the multiplication factorization theory of irreducible characters and how to define properly the characters similar to prime numbers and prime powers,or equivalently,the precise description of prime characters and primary characters.Furthermore,the research about the existence and uniqueness of prime factorizations and primary factorizations of irreducible characters,and the multiplication factorizations and tensor induction techniques for linear representations and projective representations of solvable groups founded by Berger are all deep theme in the theory of finite group representations.The research problems and results of this thesis can be regarded as a preliminary discussion in this direction.