# 1. What is Abstract Algebra | Definitions | Theorems | Examples | | ----------- | -------- | -------- | | | | | | | | | # 2. Sets & Binary Operations | Definitions | Theorems | Examples | | ----------------------------------------------------------------------------------------------------------------------------------- | -------- | ----------------------------------------------------------- | | 2.1.1 *[[Sets]] | | | | 2.2.1 [[Zermelo Frankel set theory (ZFC)]] | | | | | | | | 2.4.1 *[[Binary Operation]] | | | | 2.5 You should understand [[Vector Addition, Subtraction & Scalar Multiplication]] and their properties. Definitions not required. | | | | 2.6.1 [[Matrix]] | | | | 2.7.1 [[Ring of Polynomial Forms]] | | | | 2.8 You should understand Cayley tables and their properties. Definitions not required | | | | 2.9.1 [[Commutativity]] & [[Associativity]] | | 2.9.2.6 *[[Commutative operation that is not associative]]. | # 3. Groups | Definitions | Theorems | Examples | | ------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------ | ---------------------------- | | 3.1 [[Groups]] | | | | 3.2 You should be able to prove if an algebraic structures is a group or not. | | | | 3.3.1 *[[Groups]] | | | | 3.4.1 *[[General Linear Group]] | 3.4.2 [[The General Linear Group of Order 2 is a Group Under Matrix Multiplication]] | | | 3.5 You should understand *[[Dihedral Group D8]] and its properties such as non-abelian. Definitions not required. | | 3.5.1 *[[Dihedral Group D6]] | # 4. First Theorems & Notation | Definitions | Theorems | Examples | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------- | ----------------------------------------------------------- | | | 4.1.1 [[Uniqueness of Group Identity]]. | | | | 4.1.2 [[Uniqueness of Group Inverses]]. | | | | 4.2.1 *[[Inverse of Inverse]] | | | | 4.2.2 [[Inverse of the Product of Group Elements]]. | 4.2.4 [[Group of Exponent 2 is Abelian]]. | | You should understand that [[Integer Power of Group Element|integer powers of group elements]] are defined recursively although for positive $n$ you can simply take the view that $a^n=\underbrace{aa\cdots a}_{n\mathrm{~times}}.$ You should understand the differences and similarities between [[Multiplicative Notation]] & [[Additive Notation]]. | 4.2.6.1 [[Powers of a group element only generate other group elements]]; | | | | 4.2.6.2 [[Inverse of Power of Group Element]]. | | | | 4.2.6.3 [[Power of Power of Group Element]] & [[Product of Powers of a Group Element]] | | | | 4.2.6.4 [[Power of Product of Abelian Group Elements]]. | | # 5. The Order of An Element | Definitions | Theorems | Examples | | -------------------------------- | --------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------- | | 5.1.1 [[Order of Group Element]] | 5.1.2 [[Element of a finite group is of finite order]] | | | | 5.1.9 [[Identity is the only group element of order 1]] & [[Power of Group Element is Identity only If Order divides]] | 5.1.10 [[Order of Product of Finite Order Elements of Abelian Group divides LCM of Orders of Elements]] | # 6. Subgroups | Definitions | Theorems | Examples | | ------------------------------------------------------------------- | ----------------------------------------------------------------------- | ----------------------------------------------- | | 6.1 Subgroups of $D_8$ | | | | 6.2.1 [[Subgroup]] | | | | | 6.3.1 [[Two-Step Subgroup Test]] | 6.3.8 [[Intersection of Subgroups is Subgroup]] | | 6.4.1 [[Order of Algebraic Structure]] | | | | 6.5 You should understand [[Roots of Unity]]. Definition not given. | 6.5.1 [[Roots of Unity is a Subgroup of Unit Group of Complex Numbers]] | | | | 6.6.1 [[Trivial Subgroups are Subgroups]] | | | | | | | | | | # 7. Cyclic Groups | Definitions | Theorems | Examples | | ----------------------------------------------------------------------------------- | ------------------------------------------------------------------------------ | ---------------------------------------------------------------------- | | 7.1.1 [[Subgroup Generated by Single Element is Indeed a Subgroup]] | | | | 7.1.2 [[Cyclic Group]] | 7.1.5 [[Cyclic Groups are Abelian]] | | | | 7.1.6 [[Order of Group Element Equals Order of Subgroup Generated by Element]] | | | You should understand [[Integers modulo n]] and their operations. Not defined here. | 7.2.2 [[Integers Modulo n is a Group]] | 7.2.3 *[[Integers Modulo n is Cyclic Group]] (generated by $[1]_{n}$). | # 8. Composition as a Binary Operation | Definitions | Theorems | Examples | | -------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------- | -------- | | 8.1.1 [[Function]] | | | | 8.1.2 Two functions are equal if they have the same domain, codomain and rule: $f(x)=g(x)$ for all $x$ in the domain. | | | | 8.2.1 [[Function Composition]] | 8.2.3 [[Associativity of Function Composition]] | | | 8.2.4 [[Function Inverse]] | | | | 8.2.5 [[Injection]]; [[Surjection]] & [[Bijection]] | 8.2.6 [[Bijection iff Invertible]] | | | 8.3 You should understand a set of mappings may form a group under function composition since the operation is asociative. | | | # 9. Group of Plane Isometries | Definitions | Theorems | Examples | | ---------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------- | | 9.1. [[Euclidean Metric on Real n-Space]] | | | | 9.1.2 [[Isometry of The Plane]] | 9.1.4 [[Composition of Plane Isometries is Plane Isometry]] | | | | 9.1.5 [[Plane Isometries that Fix Origin and (1,0)]] | | | | 9.1.6 [[Plane Isometries that Fix Origin are Linear]] | | | You should understand that $O_{2}(\mathbb{R})$ is the set of isometries on $\mathbb{R}^{2}$ that fix the origin. | 9.1.7 [[Second Orthogonal Group Over The Reals is Group]] | | | You should understand that $SO_{2}(\mathbb{R})$ is the set of rotations about the origin of $\mathbb{R}^{2}.$ | 9.1.10 [[Second Special Orthogonal Group Over The Reals is Subgroup of Second Orthogonal Group Over The Reals]] | | | | 9.2.1 [[General form of Plane Isometries]] | 9.2.3 [[Orthogonal Group of the Plane is a subgroup of Euclidean Group of the Plane]]. | # 10. Symmetric Groups | Definitions | Theorems | Examples | | ------------------------------------------------------------------------------------------ | --------------------------------------------------------------- | ------------------------------------------------------------- | | 10.1 [[Symmetric Group]] | | | | | 10.2.2 [[Symmetric Group is Group]] | | | 10.3 [[Symmetric Groups of Finite Degree]] | 10.3.1 [[Number of Permutations of n Letters]] | | | 10.4.1 [[Cycle Notation]] | 10.4.2 *[[Existence of Disjoint Cycle Decomposition for Permutations of n Letters]] | | | | 10.4.8 [[Disjoint Permutations Commute]] | | | 10.5 *You should understand the relationship between dihedral groups and symmetric groups. | | | # 11. The Alternating Group | Definitions | Theorems | Examples | | ----------------------------------------------- | --------------------------------------------------------------- | ------------------------------------------------------------- | | | 11.1.1 [[k-cycles can be factored into 2-cycles]] | | | 11.2 *[[Alternating Polynomial]] | 11.2.3 *[[Transposition of Alternating Polynomial]] | | | | 11.2.4 *[[Permutation of Alternating Polynomial]] | | | | 11.2.6 *[[Parity of Permutation on n Letters is Well-Defined]]. | | | 11.2.7 [[Parity of a Permutation of n letters]] | 11.2.9 [[Alternating Group is a Subgroup of Symmetric Group]] | [[Alternating Group is a Normal Subgroup of Symmetric Group]] | | | 11.2.12 [[Order of nth Alternating Group]] | | # 12. Isomorphisms of Groups | Definitions | Theorems | Examples | | ------------------------------------------------------- | ---------------------------------------------------------------------- | ------------------------------------------------------------------ | | 12.1.1 [[Homomorphisms of groups]] | | 12.1.2 *[[Integers modulo n are Isomorphic to nth Roots of Unity]] | | 12.2 [[Direct Product of Groups]] | | | | | 12.3.1 [[Order of Element of Finite Group Divides Order of The Group]] | 12.3.2 *[[Classification of Groups of Order 4]] | | 12.4 *[[Symmetric Groups of Same Order Are Isomorphic]] | | 12.3.4.2 [[Order of k-Cycle is k]] | | | | 12.3.4.3 [[Order of Product of Disjoint Permutations]] | | | | | # 13. Group Homomorphism | Definitions | Theorems | Examples | | --------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------ | | 13.1.1 [[Homomorphisms of groups]] | | | | 13.1.2 [[Kernel of Homomorphism of Groups]] & [[Image of Homomorphism of Groups]] | | 13.1.5 *[[Homomorphism from Euclidean Group over Plane to Orthogonal Group over Plane]]. | | | 13.2.1 [[Homomorphism of Groups Preserves Identity]] | | | | 13.2.2 [[Kernel of Homomorphisms of Groups is Normal Subgroup of Domain]] & [[Image of Homomorphism of Groups is Subgroup of Codomain]]. | 13.2.3 [[Homomorphism of Groups is Injective iff its Kernel only Contains the Identity of Its Domain]] | | | | 13.2.4 *[[Domain is Isomorphic to Image of Injective Group Homomorphism]]. | # 14. Rings | Definitions | Theorems | Examples | | ------------- | ----------------------------------------- | -------- | | 14.1 [[Rings]] | | | | | 14.2.3 [[Integers modulo n is Ring]] | | | | 14.2.8 [[Ring Product With Zero is Zero]] | | | | 14.2.9 [[Product With Ring Negative]] | | # 15. Subrings | Definitions | Theorems | Examples | | ------------------------- | ----------------------------------------- | -------- | | 15.1.1 [[Subring]] | 15.1.3 [[Three Step Subring Test]] | | | 15.2.1 [[Ideal of Ring]] | 15.2.3 [[Ideal with Unity is Whole Ring]] | | | 15.2.4 [[Unit in a Ring]] | 15.2.8 [[Ideal with Unit is Whole Ring]] | | | | 15.2.10 [[Ideals of Integers]] | | | | | | # 16. The Unit Group of a Ring | Definitions | Theorems | Examples | | ------------------------------------------------------ | ------------------------------------------------------------- | -------- | | 16.1.1 [[Unit Group of Ring]] | | | | | 16.1.2 [[Unit Group of Ring is a Group]] | | | 16.2 [[Field (Algebra)]] | | | | 16.3.2 [[Unit modulo n iff coprime to n]] | | | | 16.4.1 [[Integers modulo prime is a field and integers modulo composite is not]] | | | | 16.4.3 [[Fermat's little theorem]] | | | | 16.4.5 [[Euler Totient Function]] | 16.4.10 [[Euler's totient function from prime factorisation]] | | | 16.4.9 [[Euler's theorem (Number Theory)]] | | | | | | | # 17. Factorisation in Rings | Definitions | Theorems | Examples | | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------- | ------------------------------------------------------------- | | 17.1 You understand proved [[Fundamental theorem of arithmetic\|unique factorisation theorem]] using [[Bézout's lemma]], which we proved using [[Euclid's Algorithm]] which is based on [[Division with remainder for integers\|division with remainder]]. *Note that we could have proved Bézout's identity using well-ordering principle instead.* | | | | 17.2.1 [[Degree of a Polynomial]] | 17.2.2 [[Subrings of a field is are integral domains]] | | | | 17.2.3.1 [[Degree of Product of Polynomials Over Integral Domain]] | | | | 17.2.3.2 [[Units of Ring of Polynomial Forms over Integral Domain]] | | | | 17.2.3.3 [[Ring of Polynomial Forms over Integral Domain is Integral Domain]] | | | | 17.2.3.4 [[Division with Remainder Theorem for Ring of Polynomial Forms over Fields]] | | | | 17.2.6 [[Polynomial Factor Theorem]] | | | | 17.2.8 [[Ring of Polynomial Forms over Field is a Principal Ideal Domain]] | | | 17.2.10 [[Irreducible Polynomial]] | | | | 17.2.12 [[Divisibility in Ring of Polynomial Forms]] | | | | 17.2.13 [[Relatively Prime Polynomials over Integral Domain]] | 17.2.14 [[Bézout's Identity for Ring of Polynomial Forms Over Field]] | | | | 17.2.15 [[Euclid's Lemma for Irreducible Polynomial Forms over Field]] | | | | 17.2.17 [[Ring of Polynomial Forms over Field is Unique Factorisation Domain]] | | | 17.3.1 [[Fundamental Theorem of Algebra]] | | 17.3.2.1 [[Irreducible Polynomials Over The Complex Numbers]] | | | | 17.3.2 [[Irreducible Polynomials Over The Real Numbers]] | | | 17.3.3 *[[Schönemann-Eisenstein Criterion]] | 17.3.4 *[[Irreducibility of the pth Cyclotomic Polynomial]] | # 18. Cosets and Quotient Rings | Definitions | Theorems | Examples | | ---------------------- | ---------------------------------------------------- | ---------------------------- | | 18.1.1 [[Coset]] | | | | | 18.1.5 [[Necessary Condition for Equality of Cosets]] | | | 18.2 [[Quotient Ring]] | 18.2.2 [[Quotient Ring Operations are Well-Defined]] | | | | 18.2.3 [[Quotient Ring is Ring]] | 18.2.4 [[Integers modulo n]] | | | | | # 19. Ring Homomorphism | Definitions | Theorems | Examples | | --------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------ | -------------------------------------------------------------------------------------------- | | 19.1.1 [[Homomorphism of Rings]] | | 19.1.3 [[Homomorphism from ring of polynomials over reals to the complex numbers]] | | | | 19.1.4 [[Homomorphism from ring of polynomials over complex numbers to the complex numbers]] | | 19.2.1 [[Kernel of a Homomorphism of Rings]] & [[Image of a Homomorphism of Rings]] | 19.2.2 [[Kernel of Homomorphism of Rings is an Ideal of Domain]] & [[Image of ring homomorphism is a subring]] | 19.2.10 *[[Complex Numbers are Isomorphic to Quotient Ring of Ring of Polynomials Over Reals]] |