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Group Theory

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

| | | | created: 2014-02-20 22:37:53 | modified: 2019-08-04 08:19:38 | by: bookofproofs

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.Example: Examples of Cyclic Groups

7.Proposition: Finite Order of an Element Equals Order Of Generated Group

8.Proposition: Group Homomorphisms with Cyclic Groups

9.Lemma: Cyclic Groups are Abelian

10.Lemma: Subgroups of Cyclic Groups

11.Proposition: Subgroups of Finite Cyclic Groups

12.Definition: Direct Product of Groups

13.Definition: Conjugate Elements of a Group

14.Definition: Cosets

15.Proposition: Properties of Cosets

16.Lemma: Subgroups and Their Cosets are Equipotent

17.Theorem: Order of Subgroup Divides Order of Finite Group (Lagrange)

18.Theorem: Order of Cyclic Group (Fermat's Little Theorem)

19.Symmetry Groups

20.Definition: Normal Subgroups

21.Lemma: Factor Groups

22.Lemma: Group Homomorphisms and Normal Subgroups

23.Theorem: First Isomorphism Theorem for Groups

24.Theorem: Classification of Cyclic Groups

25.Theorem: Classification of Finite Groups with the Order of a Prime Number

26.Definition: Group Operation

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Bibliography (further reading)