Group Theory

ISBN-13: 9783031213069 / Angielski / Twarda / 2023

Dinesh Khattar; Neha Agrawal

Opis

This textbook focuses on the basics and complex themes of group theory taught to senior undergraduate mathematics students across universities. The contents focus on the properties of groups, subgroups, cyclic groups, permutation groups, cosets and Lagrange’s theorem, normal subgroups and factor groups, group homomorphisms and isomorphisms, automorphisms, direct products, group actions and Sylow theorems. Pedagogical elements such as end of chapter exercises and solved problems are included to help understand abstract notions. Intermediate lemmas are also carefully designed so that they not only serve the theorems but are also valuable independently. The book is a useful reference to undergraduate and graduate students besides academics.

Spis treści

1. Group....................................................................................................... 1–58
1.1 Groups................................................................................................... 4
1.2 Cayley Table.......................................................................................... 8
1.3 Elementary Properties of Groups........................................................ 32
1.4 Dihedral Groups.................................................................................. 49
2. Finite Groups and Subgroups.............................................................. 59–98
2.1 Finite Groups....................................................................................... 59
2.2 Subgroups............................................................................................ 70
2.3 Subgroup Tests.................................................................................... 71
2.4 Special Class of Subgroups................................................................. 82
2.5 Intersection and Union of Subgroups................................................. 91
2.6 Product of Two Subgroups................................................................ 93
3. Cyclic Groups..................................................................................... 99–118
3.1 Cyclic Groups and their Properties..................................................... 99
3.2 Generators of a Cyclic Group........................................................... 102
3.3 Subgroups of Cyclic Groups............................................................. 104
4. Permutation Groups......................................................................... 119–142
4.1 Permutation of a Set.......................................................................... 119
4.2 Permutation Group of a Set.............................................................. 121
4.3 Cycle Notation................................................................................... 124
4.4 Theorems on Permutations and Cycles .......................................... 126
4.5 Even and Odd Permutations.............................................................. 134
4.6 Alternating Group of Degree n......................................................... 138
5. Cosets and Lagrange’s Theorem................................................... 143–168
5.1 Definition of Cosets and Properties of Cosets.................................. 143
5.2 Lagrange’s Theorem and its Applications........................................ 148
5.3 Application of Cosets to Permutation Groups.................................. 164
(xii)
6. Normal Subgroups and Factor Groups ........................................ 169–194
6.1 Normal Subgroup and Equivalent Conditions for a Subgroup to be
Normal............................................................................................... 169
6.2 Factor Groups.................................................................................... 180
6.3 Commutator Subgroup of a Group and its Properties...................... 187
6.4 The G/Z Theorem.............................................................................. 189
6.5 Cauchy’s Theorem for Abelian Group............................................. 191
7. Group Homomorphism and Isomorphism........................................ 195–222
7.1 Homomorphism of Groups and its Properties.................................. 195
7.2 Properties of Subgroups under Homomorphism............................... 200
7.3 Isomorphism of Groups..................................................................... 205
7.4 Some Theorems Based on Isomorphism of Groups......................... 207
8. Automorphisms ................................................................................. 223–240
8.1 Automorphism of a Group................................................................ 223
8.2 Inner Automorphisms........................................................................ 226
8.3 Theorems Based on Automorphism of a Group............................... 228
9. Direct Products............................................................................... 241–270
9.1 External Direct Product..................................................................... 241
9.2 Properties of External Direct Products............................................. 244
9.3 U(n) as External Direct Products...................................................... 249
9.4 Internal Direct Products..................................................................... 254
9.5 Fundamental Theorem of Finite Abelian Groups............................. 258
10. Group Actions.................................................................................. 271–302
10.1 Group Actions................................................................................. 271
10.2 Kernels, Orbits and Stabilizers........................................................ 275
10.3 Group acting on themselves by Conjugation.................................. 291
10.4 Conjugacy in Sn.............................................................................. 296
11. Sylow Theorems............................................................................... 303–325
11.1 p–Groups and Sylow p–subgroups.................................................. 303
11.2 Simple Groups................................................................................. 309​

O autorze

Dinesh Khattar, Ph.D., is a professor in the Department of Mathematics, Kirori Mal College, University of Delhi, India. He also served as the Principal of Kirori Mal College from 2015 to 2018. He is a topper (Gold Medalist) in his B.Sc. and M.Sc. exams. He received Dr. S. Radhakrishnan Memorial National Teachers Award 2015 for the contribution in the field of education. He has also been awarded prestigious Commonwealth Scholarship for pursuing research in UK. He is actively involved in research and presented papers in prestigious international conferences across several countries. Dr. Khattar has been a member of curriculum development committee for B.Sc. and M.Sc. programs in various universities including University of Delhi. He is also an author of many books on mathematics.

Neha Agrawal has completed her graduation and post-graduation degrees from Kirori Mal College, University of Delhi, India. She has done her M.Phil. and Ph.D. from University of Delhi. Her area of interest is nonlinear dynamical systems and chaos theory. She has been working as an assistant professor in the Department of Mathematics, Kirori Mal College since 2012. She has teaching experience of 12 years. She has published number of research papers in international journals.