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