MA268 Algebra 3 - O-Umlaut

**Canvas**: [[MA268 Algebra 3 Workspace.canvas|MA268 Algebra 3 Workspace]]. **Summary**: Normal forms of dihedral and quaternion group elements. Use of Lagrange's theorem, Cauchy's theorem, fundamental theorem of finite Abelian groups for classification of groups and fundamental theorem of group presentations of order 4, 6, 8 or any prime. First isomorphism theorems for groups, rings and modules. --- # 2. Revision | Examples | Theorems | Definitions | | -------- | -------- | ----------- | | | | | # 3. Group Cosets and Lagrange's Theorem | Examples | Theorems | Definitions | | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------- | | [[Cosets in Abelian Group]]. | III.1.3 [[Necessary Condition for Equality of Cosets]]. III.3.1 [[Subgroup and its Left Coset, Right Coset have Same Cardinality]]. | [[Coset]]. | | Cosets of unit circle, subgroup of $\mathbb{C}^*$, are all its concentric circles so $[\mathbb{C}^*:\mathbb{S}]=\infty.$ III.4.4 Order of Subgroup of Finite Group divides Order of Group. III.4.5. [[Order of Element of Finite Group Divides Order of The Group]]. [[Finite Abelian Group with Squarefree Order is Cyclic]]. | III.3.2 [[Coset Space Partitions Subgroup]]. III.4.2 [[Lagrange's theorem (on Finite Groups)]]. III.5. [[Left and Right Coset Spaces have Same Cardinality]]. | [[Coset space]]. [[Index of Subgroup]]. [[Partition of a Set]]. | # 4. Homomorphisms and Normal Subgroups ### IV.1 Normal Subgroups | Examples | Theorems | Definitions/ Lemmas | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | --------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------- | | [[Trivial Subgroups are Normal]]. [[Every Subgroup of Abelian Group is Normal]]. Scalar Matrices (SL(R)) [[Alternating Group is a Normal Subgroup of Symmetric Group]]. | IV.1.3 [[Subgroup is Normal iff Every Conjugate is Contained in Subgroup]] IV.1.8 [[Subgroup of Index 2 is Normal]]. | [[Conjugate of Subgroup]] [[Normal Subgroup]]. | ### IV.2 The quotient group | | | | | --------------------------------------------- | ------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------- | | Coset product is not well-defined in general. | [[Product of Cosets of Normal Subgroup is Well-defined]]. [[Quotient Group is Group]]. | [[Product of Cosets of Normal Subgroup]]. [[Quotient Group]]. | ### IV.3 Linear Transformations | [[Determinant is a Homomorphism]]. [[Homomorphisms of groups]]. Isomorphism of groups preserves order. [[Alternating Group has Index Two in Symmetric Group]] [[Order of nth Alternating Group]]. | [[Homomorphism of Groups Preserves Identity]]. [[Homomorphism of Groups Preserves Inverses]]. [[Homomorphism of Groups Preserves Powers]]. [[Kernel of Homomorphisms of Groups is Normal Subgroup of Domain]]. [[Image of Homomorphism of Groups is Subgroup of Codomain]]. [[Homomorphism of Groups is Injective iff its Kernel only Contains the Identity of Its Domain]]. [[First Isomorphism Theorem for Groups]]. | [[Homomorphisms of groups]]. [[Kernel of Homomorphism of Groups]]. [[Image of Homomorphism of Groups]]. | | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------- | ### IV.4 Homomorphism & Isomorphism of Groups ### IV.5. Kernel and Image ### IV.6. The First Isomorphism Theorem ### IV.7 ### IV.9 Fundamental Theorem of Finite Abelian Groups | | | | | ---------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------ | ----------------------------------------- | | [[Classification of Groups of Order 4]]. [[Abelian Groups of Order 8]]. | [[Direct Product of Groups is Group]]. [[Fundamental Theorem of Finite Abelian Groups]]. | [[Direct Product of Groups]]. | # 5. Group Presentations | Examples | Theorems | Definitions/ Lemmas | | ------------------------------------------------ | ---------------------------------------------------- | ------------------------------------------------------------------------------ | | | V.1.1 [[Normal Form of Dihedral Group Elements]]. | V.1 [[Dihedral Group]]. | | | | V.2. [[Generated Subgroup]]. V.2.1. [[Generated Subgroup is Subgroup]]. | | V.1.4. [[Group Presentation of Dihedral Group]]. | V.4.1 [[Universal Property of Group Presentations]] | V.3 [[Group Presentation]]. | | [[Quaternion Group is Not Dihedral Group]]. | V.5.2. [[Normal Form of Quaternion Group Elements]]. | | # 6. More Classification of Groups | Examples | Theorems | Definitions/ Lemmas | | -------- | ---------------------------------------------------------------- | ------------------------------------------ | | | V1.2.2 [[Group of Exponent 2 is a Power of 2nd Cyclic Group]]. | V1.2.1 [[Group of Exponent 2 is Abelian]]. | | | V1.3.1 [[Classification of Groups of Order 6]]. | | | | V1.4.1 [[Classification of Groups of Order 8]] | | # 7. Group Actions | Examples | Theorems | Definitions/ Lemmas | | ----------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | The [[Center of Group\|centre]] of a group is the set of fixed points for the conjugation action. | [[Orbit-Stabilizer Theorem]]. | [[Group action]]. [[Orbit under Group Action]]. [[Fixed Points for Group Action]]. VII.5. [[Orbits Under Group Action are Equivalence Classes]]. [[Transitive Group Action]]. [[Stabilizer under Group Action]]. VII.1.13 [[Stabilizer is Subgroup]]. When is a stabilizer the whole group | | | [[Finite Group has Elements whose Orders are Each of The Prime Factors of Group Order (Cauchy's Theorem)]]. | | | $\#G=\#\text{Cl}_{G}(h)\times \#C_{G}(h)$ | [[Group Acts on Itself by Conjugation]]. | | | | [[Conjugacy Class of An Element of nth Symmetric Group Equals The Set of All Elements with The Same Cycle Type]]. | [[Cycle Type of Permutation of Finite Set]]. | # 8. Rings | Examples | Theorems | Definitions/ Lemmas | | ----------------------------------------------------------------------------------------------- | ----------------------------------------------------- | ------------------------------------------------------------------------------------ | | | | VIII.1.5. [[Unit Group of Ring is a Group]] | | | VIII.2.2 [[Unit group of ring of Gaussian integers]]. | VIII.2.1 [[Complex Modulus is Multiplicative]]. | | VIII.3.9 [[Complex numbers]]. | | [[Homomorphism of Rings]]. VIII.3.8. [[First Isomorphism Theorem for Rings]]. | # 9. Factorisation in Integral Domains | Examples | Theorems | Defintions/Lemmas | | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[Integers modulo prime is a field and integers modulo composite is not]]. | VIII.4.5 [[Finite Integral Domains are Fields]]. VIII.4.6 [[Ring of Polynomial Forms over Integral Domain is Integral Domain]]. | [[Zero Divisor]]. [[Integral Domain]]. VIII.4.1 [[Subrings of a field is are integral domains]]. VIII.4.4 [[Cancellation Law for Integral Domains]]. | | | | [[Divisibility]]. [[Associates in Integral Domain are Necessarily Unit Multiples of Each Other]]. | | Every non-zero element of field is a unit so fields do not contain prime or irreducible elements Note that an irreducible element may not be prime. For example, $2$ is irreducible but not prime in the integral domain $\mathbb{Z}[\sqrt{ -5 }]$. | | [[Irreducible Elements of Integral Domain]]. [[Prime Elements of Integral Domain]]. [[Prime Elements of Integral Domain are Irreducible]]. | | $\mathbb{Z}[\sqrt{ -5 }]$ is not a UFD. | | [[Unique Factorisation Domain]]. Irreducibles in UFD are prime. | | IX.5.1. [[Integers form Euclidean Domain]]. IX.5.2. [[Ring of Polynomial Forms over Field is a Euclidean Domain]]. Note that the degree function isn't Euclidean on $\mathbb{Z}[X]$. IX.5.4 [[Ring of Gaussian Integers is a Euclidean Domain]]. | | [[Euclidean Domain]]. | | [[Fundamental theorem of arithmetic]]. [[Ring of Polynomial Forms over Field is Unique Factorisation Domain]]. | [[Euclidean Domain is Unique Factorisation Domain]]. | Use [[Euclidean Valuation of Proper Divisor is Strictly Less]] to prove existence. Use [[Irreducible Elements of Euclidean Domain are Prime]] (Euclid's lemma) to prove uniqueness. Use [[Bézout's Lemma for Euclidean Domains]] to prove Euclid's lemma. | # 10. Smith Normal Form | Examples | Theorems | Definitions/ Lemmas | | -------- | ------------------------------------------------------------------------------ | ------------------- | | | [[Matrices over Principal Ideal Domains are Equivalent to Smith Normal Form]]. | | # 11. Modules | Examples | Theorems | Definitions/ Lemmas | | ------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------- | --------------------------------------- | | X1.1.1 [[Vector spaces]] is a module over a field. X1.1.2 abelian group is z-module. | | [[Modules]]. [[Submodule Test]]. | | | | | | | XI.4.1 [[Fundamental Theorem of Finitely-Generated Modules over Euclidean Rings]] | |