Group/Direct/Definition

Group

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

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

is called a group when the following properties are fulfilled.

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

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