Group/Direct/Definition

Group

A set $G$ together with a special element $e\in G$ and with a binary operation

$G\times G⟶G,(g,h)⟼g\circ h$

is called a group when the following properties are fulfilled.

1. The binary operation is associative, i.e., for all $f,g,h\in G$, we have

   $(f\circ g)\circ h=f\circ (g\circ h).$

2. The element $e$ is a neutral element, i.e., for all $g\in G$, we have

   $g\circ e=g=e\circ g.$

3. For every $g\in G$, there exists an inverse element, i.e., there exists some $h\in G$ such that

   $h\circ g=g\circ h=e.$