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Part: Group Theory

In this part, we dive deeper into the theory of groups.

Table of Contents

Examples: 1

1. Definition: Generating Set of a Group
2. Definition: Group Homomorphism
3. Proposition: Additive Subgroups of Integers
4. Theorem: Construction of Groups from Commutative and Cancellative Semigroups
5. Definition: Cyclic Group, Order of an Element
6. Proposition: Finite Order of an Element Equals Order Of Generated Group
7. Proposition: Group Homomorphisms with Cyclic Groups
8. Lemma: Cyclic Groups are Abelian
9. Lemma: Subgroups of Cyclic Groups
10. Proposition: Subgroups of Finite Cyclic Groups
11. Definition: Direct Product of Groups
12. Definition: Conjugate Elements of a Group
13. Definition: Cosets
14. Proposition: Properties of Cosets
15. Lemma: Subgroups and Their Cosets are Equipotent
16. Theorem: Order of Subgroup Divides Order of Finite Group
17. Theorem: Order of Cyclic Group (Fermat's Little Theorem)
18. Chapter: Symmetry Groups
19. Definition: Normal Subgroups
20. Lemma: Factor Groups
21. Lemma: Group Homomorphisms and Normal Subgroups
22. Theorem: First Isomorphism Theorem for Groups
23. Theorem: Classification of Cyclic Groups
24. Theorem: Classification of Finite Groups with the Order of a Prime Number
25. Definition: Group Operation

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Explanations: 2

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