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

- Definition: Group
- Axiom: Axioms of Group
- Definition: Commutative (Abelian) Group
- Definition: Subgroup
- Proposition: Criteria for Subgroups
- Proposition: Unique Solvability of $a\ast x=b$ in Groups