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 $a' \in G$ with $a' \circ a = e$.

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.