Sets and groups

Inhaltsverzeichnis

• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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).
• Tables.
• List of notations.
• Answers to exercises.