1 Preliminaries

Sets 1

Subsets 2

Relations 4

Equivalence Classes 6 Mappings or Functions

Equality of Mappings 9 Composition of Mappings

Binary Compositions 13

Permutations 14 Cyclic Permutations 17

Cycles of a Permutation 18

Disjoint Permutations 20

Some Results From Number Theory 25

The Greatest Common Divisor 29

Prime Numbers 35

Composite Numbers 36 Congruences 37

Chinese Remainder Theorem

2 Groups

Subgroups 63

Cyclic Groups 78

3 Normal Subgroups, Homomorphisms, Permutation Groups

Quotient Groups 110

Homomorphisms Isomorphisms 118

Automorphisms and Conjugate

Automorphism 173

Sabgroups

Elements 187

Partition Integer 202

Theorems Direct Sylowp-subgrup 215 Couct 217 in 235 Abelian Groups 264

Actions, Solvable

Series 298

Nilpotest Groups

347

Subringi 348

of Ring

Rings 362

364
7

Rings

Subrings 347

Sum of Two Subrings 348

Characteristic of a Ring 354

Product of Rings 362 Ideals 364

Sum of Ideals 367

Product of Ideals 371

8 Homomorphisms and Embedding of Rings

Quotient Rings 380

Homomorphisms

382

Embedding of Rings 394

More on Ideals 405

Maximal Ideals

407

9 Euclidean and Factorization Domains

Euclidean Domains 429

Prime and Irreducible Elements 440 Polynomial Rings 449

Greatest Common Divisor 469 Unique Factorization Domams 471

Noctherian Rings 505

10 Vector Spaces Subspaces 514

Sum of Subspaces