  # Abstract Algebra Topics

## 3. Groups

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

3.1.1. Groups, Subgroups, and Basic Properties of Groups

3.1.2. The Order of Elements in a Group

3.1.3. Examples of Groups:

###### 3.2. Abelian Groups

3.2.1. Abelian Groups

3.2.2. Examples of Abelian Groups

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

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

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\$:

3.4.2. Transposition Permutations in \$S_n\$:

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

3.6.2. Group Isomorphisms

3.6.3. Group Automorphisms

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

3.7.1. Left and Right Cosets of Subgroups

3.7.2. Lagrange's Theorem

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

3.8.1. The Center of a Group and Centralizers

3.8.2. Normalizers

3.8.3. Normal Subgroups of a Group

3.8.4. Simple Groups

###### 3.9. Direct Products

3.9.1. The Product of Two Subgroups

3.9.2. Direct Products of Groups

3.9.3. The Internal Direct Product

3.9.4. The Fundamental Theorem of Finite Abelian Groups

###### 3.10. Quotient Groups

3.10.1. Quotient Groups

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

3.11.1. The First Group Isomorphism Theorem

3.11.2. 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

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

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

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

3.16.2. Commutators and the Derived Subgroup of a Group

3.16.3. Solvable Groups

3.16.4. Characteristic Subgroups of a Group

###### 3.17. Group Representations

3.17.1. Group Representations

3.17.2. Maschke's Theorem

3.17.4. The Character of a Group Representation

3.17.5. The Left Regular Representation

## 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. Subnormal Series in a Group

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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.