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## Group Theory

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

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

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

| | | | created: 2014-02-20 22:37:53 | modified: 2019-02-10 15:26:40 | by: bookofproofs

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