# Abstract Algebra Topics

## 1. The Set of Integers

###### 1.2. Prime and Composite Numbers, Modular Arithmetic

## 2. Types of Functions and Operations

## 3. Groups

###### 3.1. Introduction to Groups and Subgroups

**3.1.1. Groups, Subgroups, and Basic Properties of Groups**

- Groups
- Subgroups and Group Extensions
- Basic Theorems Regarding Groups
- The Cancellation Law for Groups
- Invertible Elements in Finite Groups that are Not Equal to Themselves
- The Laws of Exponents for Groups
- The Intersection and Union of Two Subgroups

**3.1.2. The Order of Elements in a Group**

- The Order of Elements in a Group
- Distinct Powers of Elements in a Group
- Basic Theorems Regarding the Order of Elements in a Group

**3.1.3. Examples of Groups:**

###### 3.2. Abelian Groups

**3.2.1. Abelian Groups**

- Abelian Groups
- Basic Theorems Regarding Abelian Groups
- The Order of a Nonabelian Group is At Least 6

**3.2.2. Examples of Abelian Groups**

###### 3.3. Symmetric Groups, Permutation Groups, and Dihedral Groups

**3.3.1. Symmetric Groups $(S_n, \circ)$**

- Functions from a Group to Itself
- The Symmetric Groups, Sn
- The Symmetric Group of a General n-Element Set

**3.3.2. Permutation Groups, $(S_X, \circ)$**

**3.3.3. Dihedral Groups, $(D_n, \circ)$**

###### 3.4. Cycles, Transpositions, and Alternating Groups

**3.4.1. Cycle Permutations in $S_n$:**

- Cycles in Permutations
- Disjoint Cycles
- Basic Theorems Regarding Disjoint Cycles
- Permutations as Products of Cycles
- Decomposition of Permutations as Products of Disjoint Cycles

**3.4.2. Transposition Permutations in $S_n$:**

- Transposition Permutations
- Basic Theorems Regarding Transpositions
- Decomposition of Permutations as Products of Transpositions
- Even and Odd Permutations as Products of Transpositions
- The Identity Permutation

**3.4.3. Alternating Groups $(A_n, \circ)$**

###### 3.5. Cyclic Groups

**3.5.1. Cyclic Groups**

###### 3.6. Group Homomorphisms, Isomorphisms, and Automorphisms

**3.6.1. Group Homomorphisms**

- Group Homomorphisms
- Basic Theorems Regarding Group Homomorphisms
- The Kernel of a Group Homomorphism
- Group Homomorphisms Review

**3.6.2. Group Isomorphisms**

- Group Isomorphisms
- Necessary Conditions for Two Groups to Be Isomorphic
- Basic Theorems Regarding Group Isomorphisms
- Preservation of the Identities and Inverses under Group Isomorphisms
- Preservation of Special Properties under Group Isomorphisms
- Cyclic Groups and their Isomorphisms
- Cayley's Group Isomorphism Theorem
- Group Isomorphisms Review

**3.6.3. Group Automorphisms**

###### 3.7. Cosets of Subgroups and Lagrange's Theorem

**3.7.1. Left and Right Cosets of Subgroups**

- Left and Right Cosets of Subgroups
- The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group
- The Number of Elements in a Left (Right) Coset
- The Index of a Subgroup

**3.7.2. Lagrange's Theorem**

###### 3.8. Centers, Centralizers, Normalizers, and Normal Subgroups

**3.8.1. The Center of a Group and Centralizers**

- The Center of a Group G, Z(G)
- The Centralizer of a Subset A of a Group G, CG(A)
- Basic Theorems Regarding Centralizers of a Subset of a Group

**3.8.2. Normalizers**

- The Normalizer of a Subset A of a Group G, NG(A)
- Basic Theorems Regarding Normalizers of a Subset of a Group

**3.8.3. Normal Subgroups of a Group**

- Normal Subgroups
- Criteria for a Subgroup to be Normal
- H is a Normal Subgroup of G IFF H is the Kernel of a Homomorphism on G
- The Intersection of Two Normal Subgroups of a Group
- Subgroups of Finite Groups with Unique Order are Normal Subgroups
- Examples of Normal Subgroups of a Group
- Inn(G) is a Normal Subgroup of Aut(G)

**3.8.4. Simple Groups**

###### 3.9. Direct Products

**3.9.1. The Product of Two Subgroups**

- The Product HK of Two Subgroups H and K of a Group G
- The Size of HK of Two Subgroups H and K of a Group G

**3.9.2. Direct Products of Groups**

- The Direct Product of Two Groups
- The Order of an Element in an External Direct Product of Two Groups
- The Direct Product of an Arbitrary Collection of Groups
- The Canonical Projections of Direct Products of Groups
- The Weak Direct Product of an Arbitrary Collection of Groups
- The Canonical Injections of Weak Direct Products of Groups

**3.9.3. The Internal Direct Product**

- The Internal Direct Product of Two Groups
- Internal Direct Products Isomorphic to External Direct Products
- External and Internal Direct Products Review

**3.9.4. The Fundamental Theorem of Finite Abelian Groups**

###### 3.11. The Group Isomorphism Theorems

**3.11.1. The First Group Isomorphism Theorem**

- The First Group Isomorphism Theorem
- An Illustrative Proof of the First Group Isomorphism Theorem
- Criterion for (G/K)/(H/K) to be Isomorphic to (G/H) When H, K are Normal Subgroups of G
- Criterion for G to be Isomorphic to H × K When H, K are Normal Subgroups of G
- Z/nZ is Isomorphic to Zn
- R^x/<-1> is Isomorphic to R^+
- GLn(R)/SL_n(R) is Isomorphic to R^x
- G/Z(G) is Isomorphic to Inn(G)

**3.11.2. The Second Group Isomorphism Theorem**

- The Intersection of a Normal Subgroup with a Subgroup is a Normal Subgroup
- The Second Group Isomorphism Theorem

**3.11.3. The Third Group Isomorphism Theorem**

###### 3.12. Free Groups on a Set

**3.12.1. Free Groups**

###### 3.13. Conjugacy

**3.13.1. Conjugacy**

- Conjugate Elements in a Group
- IFF Criterion for y to be a Conjugate of x^n
- Conjugate Subgroups of a Group
- Conjugacy Classes of a Group
- The Class Equation

**3.13.2. p-Groups and Burnside's Theorem for p-Groups**

- p-Groups
- Burnside's Theorem for p-Groups
- Every Group of Order p^2 is Abelian
- The Center Z(G) of a Nonabelian Group of Order p^3 is p

**3.13.3. Cauchy's Theorem**

**3.13.4. Examples of Conjugacy Classes**

###### 3.14. Group Actions of a Group on a Set

**3.14.1. Group Actions**

- Group Actions of a Group on a Set
- The Trivial Group Action of a Group on a Set
- The Left and Right Regular Group Actions of a Group on Itself
- A Group Action of the Symmetric Group SX on the Set X
- The Group Action of Conjugation of a Subgroup on a Group
- Faithful Group Actions of a Group on a Set

**3.14.2. Orbits and Stabilizers**

###### 3.15. The Sylow Theorems

**3.15.1. Sylow p-Subgroups and The Sylow Theorems**

###### 3.16. Solvable Groups

**3.16.1. Composition Series and the Jordan-Hölder Theorem**

- Composition Series in a Group
- The Jordan-Hölder Theorem
- The Order of a Finite Group is the Product of the Orders of its Composition Factors

**3.16.2. Commutators and the Derived Subgroup of a Group**

- The Commutator of Two Elements in a Group
- The Derived Subgroup of a Group
- Basic Theorems Regarding the Derived Subgroup of a Group

**3.16.3. Solvable Groups**

- Solvable Groups
- Composition Factor Criterion for a Finite Group to be Solvable
- nth Derived Subgroup Criterion for a Group to be Solvable
- Every p-Group is Solvable
- Every Group of Order pq is Solvable
- Every Group of Order p^2q is Solvable
- The Homomorphic Image of a Solvable Group is Solvable

**3.16.4. Characteristic Subgroups of a Group**

###### 3.17. Group Representations

**3.17.1. Group Representations**

**3.17.2. Maschke's Theorem**

- Unitary Group Representations
- Weyl's Unitarity Trick
- Maschke's Theorem
- Homomorphisms and Endomorphisms of Group Representations
- Schur's Lemma
- Algebras over C
- Division Algebras over C
- The Dual Representation of a Group Representation

**3.17.4. The Character of a Group Representation**

- Determinant Group Representations
- The Character of a Group Representation
- The Orthogonality Theorem for Characters of Irreducible Group Representations
- Corollaries to the Orthogonality Theorem for Characters of Irreducible Group Representations

**3.17.5. The Left Regular Representation**

###### 3.18. Index of Common Groups

## 4. Rings

###### 4.1. Introduction to Rings and Subrings

**4.1.1. Rings, Subrings, and Basic Properties of Rings**

**4.1.2. Units in a Ring**

**4.1.3. Examples of Rings**

###### 4.2. Commutative Rings

**4.2.1. Commutative Rings**

**4.2.2 Examples of Commutative Rings**

###### 6.2. Ideals, Quotient Rings, and Ring Homomorphisms/Isomorphisms/Automorphisms

###### 6.3. Commutative Rings, Division Rings, and Integral Domains

###### 6.4. Special Types of Integral Domains: PIDs, EDs, and UFDs

- Principal Ideals and Principal Ideal Domains (PIDs)
- The Principal Ideal Domain of Polynomials over a Field

- Units (Multiplicatively Invertible Elements) in Rings
- The Set of Units in Mnn
- The Set of Units of a Ring forms a Group under *
- Divisors of Elements in Commutative Rings
- Associates of Elements in Commutative Rings
- The Greatest Common Divisor of Elements in a Commutative Ring
- Irreducible Elements in a Commutative Ring

## 5. Fields

###### 5.1. Introduction to Fields and Subfields

###### 5.2. Polynomials over Fields

- Polynomials over a Field
- Basic Theorems Regarding Polynomials over a Field
- The Remainder Theorem for Polynomials over a Field
- Roots of Polynomials over a Field
- The Division Algorithm for Polynomials over a Field
- Ideals in the Set of Polynomials over a Field
- The Greatest Common Divisor of Polynomials over a Field
- Reducible and Irreducible Polynomials over a Field
- Congruence Classes of Polynomials Modulo p(x) over a Field
- Basic Theorems Regarding Congruence Classes of Polynomials Modulo p(x) over a Field
- Multiplicative Inverses of Congruence Classes of Polynomials Modulo p(x) over a Field
- The Field of Congruence Classes of Polynomials Modulo p(x) over a Field
- Kronecker's Field Extension Theorem
- Roots of Polynomials over Z with Rational Roots
- Primitive Polynomials over Z
- Gauss' Primitive Polynomial Lemma
- Eisenstein's Irreducible Polynomial Criterion for Polynomials over Z that are Irreducible over Q

## 6. Subnormal Series in a Group

###### 6.1. Subnormal Series

- Subnormal Series in a Group
- Equivalent Subnormal Series in a Group
- Composition Series in a Group
- The Jordan-Hölder Theorem
- Solvable Groups
- Nilpotent Groups
- Inclusion Diagram for Cyclic, Abelian, Nilpotent, and Solvable Groups
- The Commutator of Two Elements in a Group

**3.15.1. Nilpotent Groups**

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###### References

- 1. Abstract Algebra (3rd Edition) by John A. Beachy and William D. Blair.

- 2. Abstract Algebra (3rd Edition) by David S. Dummit and Richard M. Foote.