Appendix C Theory
0.1 Sets and functions
Theorem 0.1.23 Invertible is equivalent to bijective
1.1 Groups
Proposition 1.1.7 Product group
Proposition 1.1.8 Group properties
1.2 Modular arithmetic
Theorem 1.2.5 Congruence
Theorem 1.2.8 Division algorithm
Proposition 1.2.9 Least residue
Proposition 1.2.10 Modular arithmetic
Corollary 1.2.11 Ring structure of \(\Z/n\Z\)
1.4 Dihedral groups
Theorem 1.4.5 Dihedral group
1.5 Permutations
Proposition 1.5.5 Cycle arithmetic
Theorem 1.5.6 Cycle decomposition
1.6 Homomorphisms
Proposition 1.6.3 Homomorphism properties
Proposition 1.6.5 Homomorphisms from \(D_n\)
1.7 Group actions
Theorem 1.7.3 Group actions and homomorphisms
1.8 Subgroups
Theorem 1.8.10 Kernel and image
Theorem 1.8.13 Centralizer, normalizer, center
1.9 Cyclic groups
Theorem 1.9.1 Prime factorization formulas
Proposition 1.9.2 Order of group elements
Theorem 1.9.3 Isomorphic cyclic groups
Corollary 1.9.4 Isomorphic cyclic groups
Theorem 1.9.5 Cyclic group equivalence
Theorem 1.9.6 Subgroups of infinite cyclic group
Theorem 1.9.7 Subgroups of finite cyclic group
1.10 Subgroup lattice
Proposition 1.10.1 Intersection of subgroups
Proposition 1.10.3 Subgroup generated by a set
1.11 Cosets and Lagrange’s theorem
Theorem 1.11.7 Cosets and coset space
Theorem 1.11.8 Lagrange’s theorem
Corollary 1.11.11 Cardinality of subgroups
Corollary 1.11.12 Groups of prime cardinality
1.12 Quotient groups
Theorem 1.12.2 Normal subgroups
Theorem 1.12.3 Quotient groups
Corollary 1.12.6 Normality and kernels
1.13 First isomorphism theorem
Theorem 1.13.1 Quotient map: universal property
Theorem 1.13.3 First isomorphism theorem
Lemma 1.13.4 Kernel and injectivity
1.14 Isomorphism theorems: second and third
Proposition 1.14.1 Product of subgroups
Theorem 1.14.2 Second isomorphism theorem
Theorem 1.14.5 Third isomorphism theorem
Corollary 1.14.7 Quotients of quotients
1.15 Fourth isomorphism theorem
Theorem 1.15.1 Fourth isomorphism theorem
Corollary 1.15.2 Fourth isomorphism theorem: quotients
1.16 Alternating subgroup
Theorem 1.16.2 Transpositions generate \(S_n\)
Proposition 1.16.5 Sign map
Theorem 1.16.8 Permutation group representation
1.17 Group actions: stabilizer
Theorem 1.17.3 Stabilizers
Corollary 1.17.5 Action on coset space
Theorem 1.17.6 Cayley’s theorem
Theorem 1.17.7 Subgroups of prime index
1.18 Group actions: orbits
Theorem 1.18.3 Orbit-stabilizer
Theorem 1.18.6 Orbit decomposition theorem
Theorem 1.18.8 Burnside’s lemma
1.19 Class equation
Theorem 1.19.2 Conjugacy and the class equation
Theorem 1.19.6 Center of a \(p\)-group
Corollary 1.19.7 Groups of cardinality \(p^2\)
Theorem 1.19.9 Class equation of \(S_n\)
Corollary 1.19.10 Stabilizers of cycles
Theorem 1.19.12 Simplicity of \(A_n\)
1.20 Sylow theorems: intro
Theorem 1.20.9 Sylow 1
Theorem 1.20.10 Sylow 2
Corollary 1.20.11 \(p\)-Sylow subgroups
Theorem 1.20.12 Sylow 3
Corollary 1.20.15 Sylow subgroups exist
1.21 Sylow theorems: conclusion of proof
Proposition 1.21.1 Fix point of \(p\)-group action
1.22 Sylow theorems: applications
Theorem 1.22.1 Cauchy’s theorem
Theorem 1.22.2 Groups of cardinality \(pq\)
Theorem 1.22.3 Groups of cardinality \(12\)
Theorem 1.22.4 Groups of cardinality \(p^2q\)
Theorem 1.22.6 Simplicity of groups of cardinality 60
Theorem 1.22.7 Unique simple group of cardinality 60
Corollary 1.22.8 Group of rotational symmetries of the dodecahedron
1.23 Direct products
Theorem 1.23.1 Characterization of direct products
Corollary 1.23.3 Internal direct products of Sylow subgroups
Corollary 1.23.4 Sylow theorems for abelian groups
Theorem 1.23.6 Structure of finite abelian groups: elementary divisors
1.24 Semidirect products
Theorem 1.24.1 Internal semidirect products
Proposition 1.24.3 Homomorphisms from cyclic groups
Proposition 1.24.4 Automorphism groups
Proposition 1.24.5 Isomorphic semidirect products
2.1 Rings
Proposition 2.1.6 Ring properties
2.2 Subrings, units, zero divisors
Theorem 2.2.2 Matrix ring goodies
Proposition 2.2.9 Cancellation of non-zero divisors
Theorem 2.2.10 Finite integral domains
2.3 Group rings, polynomials, power series
Proposition 2.3.8 Polynomial degree properties
Corollary 2.3.10 Intregral polynomial rings
2.4 Hamilton quaternion rings
Theorem 2.4.2 Quaternion conjugation and norm
Corollary 2.4.3 \(\HH\) is a division ring
2.5 Ring homomorphisms and ideals
Proposition 2.5.3 Ring homomorphisms
Theorem 2.5.6 Ring homomorphisms from \(\Z\)
Theorem 2.5.7 Polynomial ring homomorphisms
Proposition 2.5.15 Ideals
Proposition 2.5.17 Generated ideals
Corollary 2.5.18 Generated ideals and kernels
Proposition 2.5.21 Characteristic \(p\text{:}\) first-year’s dream
Proposition 2.5.25 Homomorphisms from simple rings
2.6 Quotient rings and isomorphism theorems
Theorem 2.6.2 Quotient map: universal property
Corollary 2.6.4 Quotient ring homomorphisms
Theorem 2.6.6 Isomorphism theorems for rings
2.7 Prime and maximal ideals
Theorem 2.7.6 Prime and maximal ideals
Theorem 2.7.10 Zorn’s lemma
Theorem 2.7.12 Maximal ideals exist
2.8 Case study: polynomial rings
Theorem 2.8.1 Division algorithm for polynomials
Theorem 2.8.3 Quotients of polynomial rings
Theorem 2.8.6 Polynomial ring over a field
Theorem 2.8.10 Prime and maximal ideals of \(F[x]\)
Corollary 2.8.12 Quotient fields of \(F[x]\)
Theorem 2.8.19 Degree and distinct roots
Corollary 2.8.20 Finite multiplicative subgroups of fields
2.9 Chinese remainder theorem
Theorem 2.9.2 Chinese remainder theorem (CRT)
Corollary 2.9.3 CRT for unit groups
Proposition 2.9.4 Coprime prime ideals in PID
Theorem 2.9.5 CRT: integers
Corollary 2.9.9 Euler’s totient function
Corollary 2.9.10 Coprime ideals: monic irreducibles
2.10 Localizations and fraction fields
Proposition 2.10.4 Localization map
Theorem 2.10.6 Fraction field
Theorem 2.10.9 Mapping property of localizations
Corollary 2.10.10 Mapping property of localizations
Corollary 2.10.11 Fraction field mapping property
Proposition 2.10.14 Kernel of \(\phi^*\)
Theorem 2.10.18 Localizations and ideals
Corollary 2.10.20 Prime ideals of \(\Z[x]\)
2.11 Euclidean domains
Theorem 2.11.3 Euclidean domains are principal
Proposition 2.11.7 GCDs and LCMs
Corollary 2.11.8 GCDs and LCMs
Theorem 2.11.9 Euclidean algorithm
2.12 PIDs and UFDs
Proposition 2.12.1 Prime implies irreducible
Theorem 2.12.2 PIDs
Theorem 2.12.5 UFDs
Proposition 2.12.7 Valuation at irreducible
Proposition 2.12.9 Irreducible factorization and divisibility
Theorem 2.12.10 PID implies UFD
Lemma 2.12.11 Uniqueness of factorizations
2.13 Gaussian integers
Proposition 2.13.3 Norm and quadratic rings of integers
Corollary 2.13.4 Existence of factorizations in quadratic rings of integers
Theorem 2.13.5 Factorization of integer primes in \(\Z[i]\)
Corollary 2.13.7 Irreducible elements of \(\Z[i]\)
Corollary 2.13.9 Sums of two squares
2.14 Gauss’s lemma
Proposition 2.14.2 Primitive decompositions
Corollary 2.14.3 Gauss’s lemma
Corollary 2.14.4 Rational roots theorem
Theorem 2.14.5 Irreducible polynomials over UFDs
Theorem 2.14.6 UFD preservation in polynomial rings
2.15 Multivariate polynomial rings and monoid algebras
Theorem 2.15.1 Polynomial UFDs
Proposition 2.15.3 Monomial expansion
Theorem 2.15.4 Homomorphisms from \(R[\boldx]\)
Theorem 2.15.6 Hilbert’s basis theorem
Corollary 2.15.7 Noetherian polynomial rings
Proposition 2.15.10 Total degree
2.16 Factorization over integral domains
Theorem 2.16.2 Reduction modulo \(I\)
Theorem 2.16.7 Eisenstein’s criterion
Theorem 2.16.10 Cyclotomic polynomial: \(p\) prime
2.17 Modules
Procedure 2.17.6 Submodule verification
Proposition 2.17.8 Submodule constructions
Theorem 2.17.12 The isomorphism theorems for modules
2.18 Direct sums and free modules
Theorem 2.18.1 Mapping properties of direct sums and products
Theorem 2.18.3 Mapping property of free modules
Corollary 2.18.7 Free modules and bases
Corollary 2.18.8 Isomorphic free modules
2.19 PID-module structure theorems: statements
Theorem 2.19.2 PID-module theorem: invariant factors
Corollary 2.19.5 PID-modules theorem: free if and only if torsion free
Theorem 2.19.7 PID-module theorem: elementary divisors
Corollary 2.19.9 Finitely generated abelian groups
2.20 PID-module structure theorems: proofs
Theorem 2.20.1 PID-module \(p\)-primary components
Theorem 2.20.3 PID-module theorem: aligned bases
Theorem 2.20.8 Finitely generated modules over Noetherian ring
Proposition 2.20.9 Finite presentations over PID
Proposition 2.20.11 Standard matrices
Theorem 2.20.13 Smith normal form
Procedure 2.20.17 Smith normal form: Euclidean domains
2.21 PID-module structure theorems: Jordan canonical forms
Proposition 2.21.1 Modules over a polynomial ring
Theorem 2.21.4 Jordan decomposition
