3 Groups

A group $G$ is a set equipped with a binary operation $(g,h)\mapsto g\cdot h$, an element $e\in G$, and a unary operation $g\mapsto g^{-1}$ satisfying the following conditions.

1. (i)

   For any $g$, $h$, $k\in G$, $(g\cdot h)\cdot k=g\cdot(h\cdot k)$.

2. (ii)

   For any $g\in G$, $e\cdot g=g\cdot e=g$.

3. (iii)

   For any $g\in G$, $g\cdot g^{-1}=g^{-1}\cdot g=e$.

Futhermore, if a group $G$ satisfies the following condition, $G$ is called commutative.

1. (iv)

   For any $g$, $h\in G$, $g\cdot h=h\cdot g$.