Definition of a Group
Definition (Group)
A group is an ordered pair \( ( G, \cdot ) \) , where \( G \) is a set and \( \cdot \) is a binary operation \( G \times G \to G \), satisfying the following axioms :
- Associativity
For all \( a, b, c \in G \),
$$ (a \cdot b ) \cdot c = a \cdot ( b \cdot c ) $$ - Identity Element
There exists a special element \( e \in G \) such that for every element \( a \in G \) ,
$$ e \cdot a = a \cdot e = a $$ - Inverse Eltement
For every \( a \in G \), there exists an \( b \in G \) such that :
$$a \cdot b = b \cdot a = e $$
