Automorphisms, Derivations And Local Properties Of Given Algebraic Systems

A dynamical system, which evolves in time through the iterated application of an underlying dynamical rule[5], is a system formed by a topological space and a continuous self-map. In algebraic words, a dynamical system is a category with the feature of ordered maps. In fact, in studying dynamical system[6], either phase space, the continuous self-map with a single parameter variant or a dynamical system should be limned by algebraic structure. Therefore, it is necessary to investigate the commonly discrete algebraic system and its maps and characteristic with the basic meaning for dynamical system.Classical group, Lie algebra and finite group are the familiarly discrete algebraic systems. In this PH.D. dissertation, we made our focus on the systems given above by using the theory of the class of groups and the method of matrix.This dissertation is organized by 6 chapters.Chapter 1 is a preface which covers the meaning and background of this thesis, the relations between the algebraic systems studied and the main results in this paper.Chapter 2 describes dynamical system in the language of category[7].In each section of Chapter 3, beginning with the depiction of given algebraic systems, i.e. the standard Borel subgroup of orthogonal group O (2 m, R ) over commutative rings, the standard Borel subalgebra of Lie algebra o (2 m, R ) over commutative rings, the standard Borel subalgebra of Lie algebra of C mtype over a commutative ring, we construct some standard automorphisms respectively, including central automorphism,graph automorphism, inner automorphism, ring automorohism, extremal automorphism etc., and depict the automorphism of algebraic systems given above systematically. The main results are Theorem 3.2.24, Theorem 3.3.19, and Theorem 3.4.16.In chapter 4, we firstly introduces two kinds of algebraic systems, i.e. the parabolic subalgebras of the general linear Lie algebra gl ( n , R ) over a commutative ring, the intermediate Lie algebras between the Lie algebra of diagonal matrices and that of upper triangular matrices over a commutative ring. After constructing some standard derivations respectively, including inner derivation,central derivation, extreme derivation and permutation derivation, we give an explicit description of the derivations on the two kinds of algebraic systems given above. The main results are Theorem 4.1.17, Corollary 4.1.18 and Theorem 4.2.17. After that, the derivations of semisimple algebras and group algebras over fields are characterized. The main results are Theorem 4.3.19 and Colrollary 4.3.21.Chapter 5 defines and studies theΦ- supplement of subgroups. After introducing the concept ofΦ- supplement of subgroups and two examples, the relations between the concept ofΦ- supplement and other concepts are investigated. From then on, we give some properties of the concept and determine the structure of finite groups by mainly using some subgroups of Sylow subgroups withΦ- supplement.In section 5.3, we give some conditions of p-nilpotent groups and supersoluble groups by using maximal subgroups of Sylow subgroups withΦ- supplement. The main results are Theorem 5.3.1, Theorem 5.3.3 and Theorem 5.3.5. In section 5.4, we make research on the structure of p - nilpotent groups, p - supersoluble groups and soluble groups with subgroups of order p 2 ,p 3 havingΦ- supplement. The main results are Theorem 5.4.2, Theorem 5.4.7, Theorem 5.4.10, Theorem 5.4.11, Theorem 5.4.14, Theorem 5.4.16 and Theorem 5.4.19.Finally, chapter 6 researches into construction ofπ- closed - Sylow- tower groups and -- groups with weakly c - normal subgroups and s - semiperputable subgroups respectively, where - is a class of groups. In the first part, we determine the structures ofπ- closed - Sylow- tower groups in which some subgroups is weakly c-normal. The main results are Theorem 6.1.10 and Theorem 6.1.12. In the second part, we give some sufficient conditions under which a group belongs to a given local formation. The main results are Theorem 6.2.7, Theorem 6.2.12 and Theorem 6.2.13.