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
