Welcome guest
You're not logged in.
335 users online, thereof 0 logged in

## 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

(none)