Outline

Chapter 1 Background and Fundamentals of Mathematics

Sets, Cartesian products 1 Relations, partial orderings, Hausdorff maximality principle, 3

equivalence relations Functions, bijections, strips, solutions of equations, 5

right and left inverses, projections Notation for the logic of mathematics 13 Integers, subgroups, unique factorization 14

Chapter 2 Groups

Groups, scalar multiplication for additive groups 19 Subgroups, order, cosets 21 Normal subgroups, quotient groups, the integers mod n 25 Homomorphisms 27 Permutations, the symmetric groups 31 Product of groups 34

Chapter 3 Rings

Rings 37 Units, domains, fields 38 The integers mod n 40 Ideals and quotient rings 41 Homomorphisms 42 Polynomial rings 45 Product of rings 49 The Chinese remainder theorem 50 Characteristic 50 Boolean rings 51

Chapter 4 Matrices and Matrix Rings

Addition and multiplication of matrices, invertible matrices 53 Transpose 56 Triangular, diagonal, and scalar matrices 56 Elementary operations and elementary matrices 57 Systems of equations 59 Determinants, the classical adjoint 60 Similarity, trace, and characteristic polynomial 64

Chapter 5 Linear Algebra

Modules, submodules 68 Homomorphisms 69 Homomorphisms on R^{n} 71 Cosets and quotient modules 74 Products and coproducts 75 Summands 77 Independence, generating sets, and free basis 78 Characterization of free modules 79 Uniqueness of dimension 82 Change of basis 83 Vector spaces, square matrices over fields, rank of a matrix 85 Geometric interpretation of determinant 90 Linear functions approximate differentiable functions locally 91 The transpose principle 92 Nilpotent homomorphisms 93 Eigenvalues, characteristic roots 95 Jordan canonical form 96 Inner product spaces, Gram-Schmidt orthonormalization 98 Orthogonal matrices, the orthogonal group 102 Diagonalization of symmetric matrices 103

Chapter 6 Appendix

The Chinese remainder theorem 108 Prime and maximal ideals and UFD^{s} 109 Splitting short exact sequences 114 Euclidean domains 116 Jordan blocks 122 Jordan canonical form 123 Determinants 128 Dual spaces 130

1 2 3 4
5 6 7 8
9 11 10

Abstract algebra is not only a major subject of science, but it is also magic and fun. Abstract algebra is not all work and no play, and it is certainly not a dull boy. See, for example, the neat card trick on page 18. This trick is based, not on sleight of hand, but rather on a theorem in abstract algebra. Anyone can do it, but to understand it you need some group theory. And before beginning the course, you might first try your skills on the famous (some would say infamous) tile puzzle. In this puzzle, a frame has 12 spaces, the first 11 with numbered tiles and the last vacant. The last two tiles are out of order. Is it possible to slide the tiles around to get them all in order, and end again with the last space vacant? After giving up on this, you can study permutation groups and learn the answer!