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Group Theory

Above, we have learned about magmas, semigroups, monoids as simple types of algebraic structures. In this chapter, we will introduce a more complex algebra – the group. We continue with our tabular overview to indicate, which properties of a group fulfills:

Algebra $(X,\ast)$ Closure Associativity Neutral Element Existence of Inverse Cancellation Commutativity
Magma required (✔) (✔) (✔) (✔) (✔)
Semigroup required required (✔) (✔) (✔) (✔)
Monoid required required required (✔) (✔) (✔)
Group required required required required (✔)

We will see later that, in every group, the existence of inverse elements already ensures the cancellation property. Therfore the entry is “✔” and not the optional “(✔)”.

Groups are so important structures in algebra that group theory can be concidered as a separate branch of algebra and mathematics. The results of group theory have various applications in physics and technology.

| | | | created: 2014-02-20 22:37:53 | modified: 2019-02-10 15:26:40 | by: bookofproofs

1.Definition: Group

2.Definition: Group Order

3.Definition: Commutative (Abelian) Group

4.Motivation: Calculations in a Group

5.Example: Examples of Groups

6.Definition: Subgroup

7.Theorem: Construction of Groups from Commutative and Cancellative Semigroups

8.Symmetry Groups

9.Definition: Group Homomorphism

10.Proposition: Properties of Cosets

11.Definition: Direct Product of Groups

12.Definition: Cosets

13.Definition: Cyclic Group

14.Definition: Group Operation


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