Fundamentals of Group Theory [recurso electrónico] An Advanced Approach

Fundamentals of Group Theory provides a comprehensive account of the basic theory of groups. Both classic and unique topics in the field are covered, such as an historical look at how Galois viewed groups, a discussion of commutator and Sylow subgroups, and a presentation of Birkhoff’s theorem. Written in a clear and accessible style, the work presents a solid introduction for students wishing to learn more about this widely applicable subject area. This book will be suitable for graduate courses in group theory and abstract algebra, and will also have appeal to advanced undergraduates. In addition it will serve as a valuable resource for those pursuing independent study. Group Theory is a timely and fundamental addition to literature in the study of groups. Fundamentals Of Group Theory Provides An Advanced Look At The Basic Theory Of Groups. Standard Topics In The Field Are Covered Alongside A Great Deal Of Unique Content. There Is An Emphasis On Universality When Discussing The Isomorphism Theorems, Quotient Groups And Free Groups As Well As A Focus On The Role Of Applying Certain Operations, Such As Intersection, Lifting And Quotient To A “group Extension”. Certain Concepts, Such As Subnormality, Group Actions And Chain Conditions Are Introduced Perhaps A Bit Earlier Than In Other Texts At This Level, In The Hopes That The Reader Would Acclimate To These Concepts Earlier. Some Additional Features Of The Work Include: An Historical Look At How Galois Viewed Groups. The Problem Of Whether The Commutator Subgroup Of A Group Is The Same As The Set Of Commutators Of The Group, Including An Example Of When This Is Not The Case.^ The Subnormal Join Property, That Is, The Property That The Join Of Two Subnormal Subgroups Is Subnormal. Cancellation In Direct Sums. A Complete Proof Of The Theorem Of Baer Characterizing Nonabelian Groups With The Property That All Of Their Subgroups Are Normal. A Somewhat More In Depth Discussion Of The Structure Of P-groups, Including The Nature Of Conjugates In A P-group, A Proof That A P-group With A Unique Subgroup Of Any Order Must Be Either Cyclic (for P>2) Or Else Cyclic Or Generalized Quaternion (for P=2) And The Nature Of Groups Of Order P N That Have Elements Of Order P (n-1). A Discussion Of The Sylow Subgroups Of The Symmetric Group In Terms Of Wreath Products. An Introduction To The Techniques Used To Characterize Finite Simple Groups. Birkhoff's Theorem On Equational Classes And Relative Freeness. This Book Is Suitable For A Graduate Course In Group Theory, Part Of A Graduate Course In Abstract Algebra Or For Independent Study.^ It Can Also Be Read By Advanced Undergraduates. The Book Assumes No Specific Background In Group Theory, But Does Assume Some Level Of Mathematical Sophistication On The Part Of The Reader. 1. Preliminaries -- 2. Groups And Subgroups -- 3. Cosets, Index And Normal Subgroups -- 4. Homomorphisms, Chain Conditions And Subnormality -- 5. Direct And Semidirect Products -- 6. Permutation Groups -- 7. Group Actions; The Structure Of P-groups -- 8. Slow Theory -- 9. The Classification Problem For Groups -- 10. Finiteness Conditions -- 11. Solvable And Nilpotent Groups -- 12. Free Groups And Presentations -- 13. Abelian Groups. Steven Roman. Includes Bibliographical References (p. 367-369) And Index. Fundamentals of Group Theory provides an advanced look at the basic theory of groups. Standard topics in the field are covered alongside a great deal of unique content. There is an emphasis on universality when discussing the isomorphism theorems, quotient groups and free groups as well as a focus on the role of applying certain operations, such as intersection, lifting and quotient to a "group extension". Certain concepts, such as subnormality, group actions and chain conditions are introduced perhaps a bit earlier than in other texts at this level, in the hopes that the reader would acclimate to these concepts earlier. Some additional features of the work include: An historical look at how Galois viewed groups. The problem of whether the commutator subgroup of a group is the same as the set of commutators of the group, including an example of when this is not the case. The subnormal join property, that is, the property that the join of two subnormal subgroups is subnormal. Cancellation in direct sums. A complete proof of the theorem of Baer characterizing nonabelian groups with the property that all of their subgroups are normal. A somewhat more in depth discussion of the structure of p-groups, including the nature of conjugates in a p-group, a proof that a p-group with a unique subgroup of any order must be either cyclic (for p>2) or else cyclic or generalized quaternion (for p=2) and the nature of gro ups of order p^n that have elements of order p^(n-1). A discussion of the Sylow subgroups of the symmetric group in terms of wreath products. An introduction to the techniques used to characterize finite simple groups. Birkhoff's theorem on equational classes and relative freeness. This book is suitable for a graduate course in group theory, part of a graduate course in abstract algebra or for independent study. It can also be read by advanced undergraduates. The book assumes no specific background in group theory, but does assume some level of mathematical sophistication on the part of the reader Fundamentals of Group Theory provides an advanced look at the basic theory of groups. Standard topics in the field are covered alongside a great deal of unique content. There is an emphasis on universality when discussing the isomorphism theorems, quotient groups and free groups as well as a focus on the role of applying certain operations, such as intersection, lifting and quotient to a "group extension". Certain concepts, such as subnormality, group actions and chain conditions are introduced perhaps a bit earlier than in other texts at this level, in the hopes that the reader would acclimate to these concepts earlier. Some additional features of the work include: An historical look at how Galois viewed groups.The problem of whether the commutator subgroup of a group is the same as the set of commutators of the group, including an example of when this is not the case.The subnormal join property, that is, the property that the join of two subnormal subgroups is subnormal.Cancellation in direct sums.A complete proof of the theorem of Baer characterizing nonabelian groups with the property that all of their subgroups are normal.A somewhat more in depth discussion of the structure of p-groups, including the nature of conjugates in a p-group, a proof that a p-group with a unique subgroup of any order must be either cyclic (for p>2) or else cyclic or generalized quaternion (for p=2) and the nature of groups of order p^n that have elements of order p^(n-1).A discussion of the Sylow subgroups of the symmetric group in terms of wreath products.An introduction to the techniques used to characterize finite simple groups. Birkhoff's theorem on equational classes and relative freeness.This book is suitable for a graduate course in group theory, part of a graduate course in abstract algebra or for independent study. It can also be read by advanced undergraduates. The book assumes no specific background in group theory, but does assume some level of mathematical sophist ication on the part of the reader __Fundamentals of Group Theory__ provides a comprehensive account of the basic theory of groups. Both classic and unique topics in the field are covered, such as an historical look at how Galois viewed groups, a discussion of commutator and Sylow subgroups, and a presentation of Birkhoff’s theorem. Written in a clear and accessible style, the work presents a solid introduction for students wishing to learn more about this widely applicable subject area. This book will be suitable for graduate courses in group theory and abstract algebra, and will also have appeal to advanced undergraduates. In addition it will serve as a valuable resource for those pursuing independent study. __Group Theory__ is a timely and fundamental addition to literature in the study of groups. Front Matter....Pages i-xii Preliminaries....Pages 1-17 Groups and Subgroups....Pages 19-60 Cosets, Index and Normal Subgroups....Pages 61-103 Homomorphisms, Chain Conditions and Subnormality....Pages 105-148 Direct and Semidirect Products....Pages 149-190 Permutation Groups....Pages 191-206 Group Actions; The Structure of P -Groups....Pages 207-233 Sylow Theory....Pages 235-262 The Classification Problem for Groups....Pages 263-271 Finiteness Conditions....Pages 273-289 Solvable and Nilpotent Groups....Pages 291-317 Free Groups and Presentations....Pages 319-352 Abelian Groups....Pages 353-365 Back Matter....Pages 367-380 Preliminaries.- Groups and Subgroups.- Cosets, Index and Normal Subgroups.- Homomorphisms.- Chain Conditions and Subnormality.- Direct and Semidirect Products.- Permutation Groups.- Group Actions.- The Structure of -Groups.- Sylow Theory.- The Classification Problem for Groups.- Finiteness Conditions.- Free Groups and Presentations.- Abelian Groups.- References