Sets and groups

1 Sets.- 1.1 Sets.- 1.2 Subsets.- 1.3 Intersection.- 1.4 Union.- 1.5 The algebra of sets.- 1.6 Difference and complement.- 1.7 Pairs. Product of sets.- 1.8 Sets of sets.- Exercises.- 2 Equivalence relations.- 2.1 Relations on a set.- 2.2 Equivalence relations.- 2.3 Partitions.- 2.4 Equivalence classes.- 2.5 Congruence of integers.- 2.6 Algebra of congruences.- Exercises.- 3 Maps.- 3.1 Maps.- 3.2 Equality of maps.- 3.3 Injective, surjective, bijective maps. Inverse maps.- 3.4 Product of maps.- 3.5 Identity maps.- 3.6 Products of bijective maps.- 3.7 Permutations.- 3.8 Similar sets.- Exercises.- 4 Groups.- 4.1 Binary operations on a set.- 4.2 Commutative and associative operations.- 4.3 Units and zeros.- 4.4 Gruppoids, semigroups and groups.- 4.5 Examples of groups.- 4.6 Elementary theorems on groups.- Exercises.- 5 Subgroups.- 5.1 Subsets closed to an operation.- 5.2 Subgroups.- 5.3 Subgroup generated by a subset.- 5.4 Cyclic groups.- 5.5 Groups acting on sets.- 5.6 Stabilizers.- Exercises.- 6 Cosets.- 6.1 The quotient sets of a subgroup.- 6.2 Maps of quotient sets.- 6.3 Index. Transversals.- 6.4 Lagrange's theorem.- 6.5 Orbits and stabilizers.- 6.6 Conjugacy classes. Centre of a group.- 6.7 Normal subgroups.- 6.8 Quotient groups.- Exercises.- 7 Homomorphisms.- 7.1 Homomorphisms.- 7.2 Some lemmas on homomorphisms.- 7.3 Isomorphism.- 7.4 Kernel and image.- 7.5 Lattice diagrams.- 7.6 Homomorphisms and subgroups.- 7.7 The second isomorphism theorem.- 7.8 Direct products and direct sums of groups.- Exercises.- 8 Rings and fields.- 8.1 Definition of a ring. Examples.- 8.2 Elementary theorems of rings. Subrings.- 8.3 Integral domains.- 8.4 Fields. Division rings.- 8.5 Polynomials.- 8.6 Homomorphisms. Isomorphism of rings.- 8.7 Ideals.- 8.8 Quotient rings.- 8.9 The Homomorphism Theorem for rings.- 8.10 Principal ideals in a commutative ring.- 8.11 The Division Theorem for polynomials.- 8.12 Polynomials over a field.- 8.13 Divisibility in Z and in F[X].- 8.14 Euclid's algorithm.- Exercises.- 9 Vector spaces and matrices.- 9.1 Vector spaces over a field.- 9.2 Examples of vector spaces.- 9.3 Two geometric interpretations of vectors.- 9.4 Subspaces.- 9.5 Linear combinations. Spanning sets.- 9.6 Linear dependence. Basis of a vector space.- 9.7 The Basis Theorem. Dimension.- 9.8 Linear maps. Isomorphism of vector spaces.- 9.9 Matrices.- 9.10 Laws of matrix algebra. The ring Mn(F).- 9.11 Row space of a matrix. Echelon matrices.- 9.12 Systems of linear equations.- 9.13 Matrices and linear maps.- 9.14 Invertible matrices. The group GLn(F).- Exercises.- Tables.- List of notations.- Answers to exercises.