Definition Groups

A group is a pair $(G,\circ )$ consisting of a set $G$ and an internal operation $\circ :G\times G\to G$, $(a,b)\mapsto a\circ b$, with the properties

G1	Associative law: $(a\circ b)\circ c=a\circ (b\circ c)$ for all $a,b,c\in G$.
G2	Neutral element: There exists an $e\in G$ such that $e\circ a=a$ for all $a\in G$.
G3	Inverse element: For all $a\in G$, there exists an $a^{\prime }\in G$ with $a^{\prime }\circ a=e$.

If, in addition, the commutative law,

$$

a \circ b = b \circ a \quad \text{for all } a, b \in G,

$$

holds, then the group is called abelian.