Abstract Algebra in GAP Alexander Hulpke with contributions by Kenneth Monks and Ellen Ziliak Version of January 2011 Alexander Hulpke Department of Mathematics Colorado State University 1874 Campus Delivery Fort Collins, CO, 80523 c 2008-2011 by the authors. This work is licensed under the Creative Commons Attribution- Noncommercial-Share Alike 3.0 United States License. To view a copy of this license, visit http: //creativecommons.org/licenses/by-nc-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. - Contents Contents 1 1 The Basics 7 1.1 Installation . 7 Windows . 7 Macintosh . 7 Linux ............................................ 8 1.2 The GGAP user interface . 8 Workspaces and Worksheets . 8 1.3 Basic System Interaction . 9 Comparisons . 9 Variables and Assignment . 10 Functions . 10 Lists . 11 Vectors and Matrices . 13 Sets and sorted lists . 14 List operations . 14 Basic Programming . 16 1.4 File Operations . 17 Files and Directories . 17 Working directories . 18 Input and Output . 18 File Input . 19 1.5 Renaming Objects . 20 1.6 Teaching Mode . 20 1.7 Handouts . 21 Introduction to GAP .................................... 21 1.8 Problems . 31 2 Rings 33 2.1 Integers, Rationals and Complex Numbers . 33 Rationals . 33 1 Integers . 33 2.2 Complex Numbers and Cyclotomics . 34 Quadratic Extensions of the Rationals . 35 2.3 Greatest common divisors and Factorization . 35 2.4 Modulo Arithmetic . 36 Checksum Digits . 36 2.5 Residue Class Rings and Finite Fields . 38 2.6 Arithmetic Tables . 39 2.7 Polynomials . 40 Polynomial rings . 41 Operations for polynomials . 41 2.8 Small Rings . 43 Creating Small Rings . 45 Creating rings from arithmetic tables . 47 2.9 Ideals, Quotient rings, and Homomorphisms . 47 2.10 Resultants and Groebner Bases . 47 2.11 Handouts . 51 Some aspects of Gr¨obnerbases . 51 Reed Solomon Codes . 55 2.12 Problems . 61 3 Groups 67 3.1 Cyclic groups, Abelian Groups, and Dihedral Groups . 67 3.2 Units Modulo . 68 3.3 Groups from arbitrary named objects . 69 3.4 Basic operations with groups and their elements . 69 3.5 Left and Right . 70 3.6 Groups as Symmetries . 71 3.7 Multiplication Tables . 71 3.8 Generators . 72 3.9 Permutations . 73 3.10 Matrices . 73 3.11 Finitely Presented Groups . 74 3.12 Subgroups . 75 3.13 Subgroup Lattice . 75 3.14 Group Actions . 78 3.15 Group Homomorphisms . 80 Action homomorphisms . 80 Using generators . 80 By prescribing images with a function . 81 Operations for group homomorphisms . 81 3.16 Factorization . 82 3.17 Cosets . 83 3.18 Factor Groups . 84 3.19 Finding Symmetries . 85 3.20 Identifying groups . 87 2 3.21 Pc groups . 89 3.22 Special functions for the book by Gallian . 89 3.23 Handouts . 91 Permutations . 91 Groups generated by elements . 93 Solving Rubik's Cube by hand . 98 The subgroups of S5 ....................................101 3.24 Problems . 102 4 Linear Algebra 107 4.1 Operations for matrices . 107 Creating particular Matrices . 108 Eigenvector theory . 109 Normal Forms . 110 4.2 Problems . 111 5 Fields and Galois Theory 113 5.1 Field Extensions . 113 Polynomials for iterated extensions . 114 5.2 Galois groups over the rationals . 117 5.3 Handouts . 118 Identification of a Galois group by Cycle Shapes . 118 5.4 Problems . 120 6 Number Theory 123 6.1 Units.