Chapter: 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.

Motivations: 1 Examples: 1 Explanations: 1

  1. Definition: Group
  2. Axiom: Axioms of Group
  3. Definition: Commutative (Abelian) Group
  4. Definition: Subgroup
  5. Proposition: Criteria for Subgroups
  6. Proposition: Unique Solvability of $a\ast x=b$ in Groups

Chapters: 1
Parts: 2

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