Groups

Table of contents

1. Definition
   1. Closure
   2. Associativity
   3. Existence of a Neutral Element
   4. Existence of an Inverse Element
2. Abelian Group
   1. Commutativity

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Definition

Let $\mathcal{G}$ be a set and $\otimes: \mathcal{G} \times \mathcal{G} \to \mathcal{G}$ an inner operation on $\mathcal{G}$.

Then $G = (\mathcal{G}, \otimes)$ is a group if it satisfies the following conditions:

Closure

Closure of $\mathcal{G}$ under $\otimes$:

$$ \forall x,y \in \mathcal{G},\, x \otimes y \in \mathcal{G} $$

Associativity

$$ \forall x,y,z \in \mathcal{G},\, (x \otimes y) \otimes z = x \otimes (y \otimes z) $$

Existence of a Neutral Element

$$ \exists e \in \mathcal{G} \mathbin{s.t.} \forall x \in \mathcal{G},\, x \otimes e = x \wedge e \otimes x = x $$

Existence of an Inverse Element

$$ \forall x \in \mathcal{G},\, \exists y \in \mathcal{G} \mathbin{s.t.} x \otimes y = e \wedge y \otimes x = e $$

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Abelian Group

If a group additionally satisfies the following property, it is called an Abelian group:

Commutativity

$$ \forall x,y \in \mathcal{G},\, x \otimes y = y \otimes x $$