Group/Direct/Definition

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

${\displaystyle G\times G\longrightarrow G,(g,h)\longmapsto g\circ h}$

is called a group when the following properties are fulfilled.

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

   ${\displaystyle {}(f\circ g)\circ h=f\circ (g\circ h)\,.}$

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

   ${\displaystyle {}g\circ e=g=e\circ g\,.}$

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

   ${\displaystyle {}h\circ g=g\circ h=e\,.}$