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Groups (Overview)

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 (✔) (✔) (✔) (✔) (✔)
Semigroup (✔) (✔) (✔) (✔)
Monoid (✔) (✔) (✔)
Group (✔)

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

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

| | | | created: 2019-08-09 19:25:51 | modified: 2019-08-09 19:25:51 | by: bookofproofs

1.Motivation: Calculations in a Group

2.Definition: Group

3.Definition: Group Order

4.Definition: Commutative (Abelian) Group

5.Example: Examples of Groups

6.Definition: Subgroup

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Bibliography (further reading)