Distances in Algebra

Abstract

A group (G, ⋅ , e) is a set G of elements with a binary operation ⋅ , called the group operation, that together satisfy the four fundamental properties of closure (x ⋅ y ∈ G for any x, y ∈ G), associativity (x ⋅ (y ⋅ z) = (x ⋅ y) ⋅ z for any x, y, z ∈ G), the identity property ($x \cdot e = e \cdot x = x$ for any x ∈ G), and the inverse property (for any x ∈ G, there exists an element x ⁻¹ ∈ G such that $x \cdot x^{-1} = x^{-1} \cdot x = e$).