What Are Group Axioms?

A group is a set G such that the following four requirements, known as group axioms, are satisfied.

1. Closure property
2. Associativity
3. Identity element
4. Inverse element

We will look further for each of them with examples.

1. Closure property

Let the symbol * be a binary operation on a set G. A binary operation means the calculation that combines two elements to produce another element. For example, addition 1 + 2 = 3 is a binary operation because it combines two elements, 1 and 2, to produce another element, 3. And the symbol * can be any binary operations including subtraction(−), multiplication(×), etc.

We will write performing the operation * on a and b as “a * b.” And if a * b is also an element of G for any two elements a and b in G, we call it “a set G is closed under the operation * .” And this property is called closure property.

For simplicity, we introduce the symbol “∈”;

a ∈ G
means a is an element of G. Similarly,
a * b ∈ G
means a * b is an element of G.

One of the examples that satisfies the closure property is the set of all the natural numbers ℕ = {1, 2, 3, ……} under addition(+).

Suppose
• a = 2
• b = 5
and of course
• a = 2 ∈ ℕ
• b = 5 ∈ ℕ
then
a + b = 2 + 5 = 7 ∈ ℕ