Group Axioms

A group is a non-empty set \(G\) together with a binary operation on \(G\) (\(\cdot\)), that fulfills the following axioms:
1. Associativity: For all \(a, b, c \in G\), one has \((a \cdot b) \cdot c = a \cdot (b \cdot c)\)
2. Identity element: There exists an element \(e\in G\) such that, for every \(a \in G\), \(e \cdot a = a\) and \(a \cdot e = a\)
3. Inverse element: For each \(a\in G\), there exists a unique element \(b\in G\) such that \(a \cdot b = e\) and \(b \cdot a = e\), where \(e\) is the identity element. The inverse of \(a\) is denoted as \(a^{-1}\)