Group

A group is a set $G$ equipped with a binary operation $*:G\times G \rightarrow G$ satisfying the following properties :

1. Associativity : for all $a,b,c\in G$, $(a*b)*c=a*(b*c)$.
2. Identity element : there exists an element $e\in G$ such that $e*a=a*e=a$ for every $a\in G$.
3. Inverses : for every $a\in G$, there exists an element $a^{-1}\in G$ such that $a*a^{-1}=a^{-1}*a=e$.

Remarks

• When the context is clear, the operation $*$ is often ommited, writing $ab$ for $a*b$.