Groups

Table of contents

Definition

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

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

Closure of $G$ under $\otimes$ :

$\forall x, y \in G, x \otimes y \in G$

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

$\exists e \in G s . t . \forall x \in G, x \otimes e = x \land e \otimes x = x$

$\forall x \in G, \exists y \in G s . t . x \otimes y = e \land y \otimes x = e$

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

$\forall x, y \in G, x \otimes y = y \otimes x$