Appendix D Examples
0.1 Sets and functions
Example 0.1.11 Power set
Example 0.1.14 Arithmetic operations as functions
Example 0.1.20 Role of domain and codomain in injectivity and surjectivity
0.2 Logic
Example 0.2.7 Modeling “Every positive number has a square-root”
Example 0.2.10 The limit does not exist
0.3 Proof techniques
Example 0.3.3 Proof: invertible is equivalent to bijective
Example 0.3.4 Proof by contradiction
Example 0.3.8 Weak induction
Example 0.3.10 Strong induction
1.1 Groups
Example 1.1.4 \(\GL_n(R)\) is nonabelian
Example 1.1.11 Order of inverse
Example 1.1.12 Element of infinite order
1.2 Modular arithmetic
Example 1.2.3 Congruence modulo \(3\)
Example 1.2.12 Modular arithmetic
Example 1.2.13 Least residue
Example 1.2.15 Modulus \(n=5\)
Example 1.2.16 Units modulo \(9\)
1.3 Matrix groups
Example 1.3.3 \(\GL_2(\Z/2\Z)\)
1.4 Dihedral groups
Example 1.4.2 Isometries
Example 1.4.6 Computing in \(D_3\)
Example 1.4.7 Dihedral geometry
1.5 Permutations
Example 1.5.3 Permutations: table notation
Example 1.5.8 Cycle decomposition
1.6 Homomorphisms
Example 1.6.4 Homormorphisms
Example 1.6.7 Permutation groups
1.7 Group actions
Example 1.7.2 Group actions
Example 1.7.5 Isomorphisms
1.8 Subgroups
Example 1.8.4 Examples
Example 1.8.6 Cyclic groups
Example 1.8.8 Subgroups of \(D_n\)
Example 1.8.11 Kernel and image
Example 1.8.14 Centralizer, normalizer, center
1.11 Cosets and Lagrange’s theorem
Example 1.11.4 Cosets in \(\Z\)
Example 1.11.5 Cosets in \(\Z/12\Z\)
Example 1.11.6 Cosets in \(D_4\)
1.12 Quotient groups
Example 1.12.4 Normality in \(D_4\)
Example 1.12.5 Abelian groups
Example 1.12.7 Normality in \(S_4\)
1.13 First isomorphism theorem
Example 1.13.5 First isomorphism theorem
Example 1.13.6 First isomorphism theorem: \(\SL_n(R)\)
Example 1.13.7 First isomorphism theorem: \(\R/\Z\)
1.16 Alternating subgroup
Example 1.16.4 Sign of permutations
Example 1.16.6 \(A_4\)
1.18 Group actions: orbits
Example 1.18.4 Rotational symmetries of a tetrahedron
Example 1.18.5 Rotational symmetries of a tetrahedron
1.19 Class equation
Example 1.19.4 Class equation: abelian groups
Example 1.19.5 Class equation: \(D_8\)
Example 1.19.11 Class equation: \(A_5\)
1.20 Sylow theorems: intro
Example 1.20.4 Sylow subgroups of \(S_3\)
Example 1.20.5 Sylow subgroups of cyclic groups
Example 1.20.6 Sylow subgroups of \(A_4\)
Example 1.20.7 Sylow subgroups of \(S_4\)
1.22 Sylow theorems: applications
Example 1.22.5 Groups of cardinality 20
1.24 Semidirect products
Example 1.24.6 Groups of cardinality 21
Example 1.24.7 Groups of cardinality \(12\)
Example 1.24.8 Groups of cardinality \(28\)
2.1 Rings
Example 2.1.4 Trivial ring
Example 2.1.9 Center of a ring
2.2 Subrings, units, zero divisors
Example 2.2.3 Units in product rings
Example 2.2.5 Division rings
Example 2.2.8 Zero divisors
2.3 Group rings, polynomials, power series
Example 2.3.3 Group ring of \(D_3\)
Example 2.3.5 Group ring properties
Example 2.3.9 Polynomial degree
Example 2.3.11 Units of polynomial and power series rings
2.5 Ring homomorphisms and ideals
Example 2.5.5 \(\Hom(\Z/3\Z, \Z/2\Z)\)
Example 2.5.8 \(\Hom(\Z[x],R)\)
Example 2.5.12 Augmentation map
Example 2.5.14 Trivial ideals
Example 2.5.19 Ideals of \(\Z\)
Example 2.5.22 Ideals of division rings
Example 2.5.23 Ideals of \(M_n(R)\)
2.6 Quotient rings and isomorphism theorems
Example 2.6.1 Modular rings
Example 2.6.5 Quotient ring homomorphisms
Example 2.6.7 First isomorphism theorem
Example 2.6.8 Ideals of \(\Z/n\Z\)
Example 2.6.9 Quotients of \(\Z/n\Z\)
2.7 Prime and maximal ideals
Example 2.7.2 Division ring
Example 2.7.3 Maximal ideals of \(M_n(F)\)
Example 2.7.4 Maximal ideals of \(\Z\)
Example 2.7.7 Prime and maximal ideals of \(\Z\)
2.8 Case study: polynomial rings
Example 2.8.14 Quotients by linear polynomials
Example 2.8.15 Quotients of \(\R[x]\)
Example 2.8.16 Field of cardinality 25
2.9 Chinese remainder theorem
Example 2.9.7 CRT for integers
Example 2.9.11 CRT in polynomial rings
2.10 Localizations and fraction fields
Example 2.10.8 Fraction fields
Example 2.10.16 Localizing at an element: \(\Z_2\)
Example 2.10.17 Localization of \(\Z[x]\)
2.11 Euclidean domains
Example 2.11.10 GCD in polynomial ring
Example 2.11.11 Ideal generator in \(\Z[i]\)
2.12 PIDs and UFDs
Example 2.12.3 \(\Z[\sqrt{-5}]\) is not a PID
2.15 Multivariate polynomial rings and monoid algebras
Example 2.15.8 Prime ideal in \(R[x,y]\)
Example 2.15.11 Failure of division algorithm
Example 2.15.13 Monoid examples
2.16 Factorization over integral domains
Example 2.16.5 Reduction mod \(I\text{:}\) cubic
Example 2.16.6 Reduction mod \(I\text{:}\) quartic
Example 2.16.8 Eisenstein
Example 2.16.9 Eisenstein: \(n\)-th roots in UFD
2.17 Modules
Example 2.17.2 Ring objects as modules
Example 2.17.3 Modules over fields
Example 2.17.4 Modules over \(\Z\)
2.18 Direct sums and free modules
Example 2.18.5 Finitely generated modules
Example 2.18.9 Homomorphisms between free modules
2.20 PID-module structure theorems: proofs
Example 2.20.2 \(p\)-primary components
Example 2.20.19 Aligned bases
Example 2.20.20 Computing a cokernel
