Axioms Of A Group

Question 1

What are the axioms of a group?

Answer 1

1. Uniqueness of the identity element
2. Cancellation laws
3. Existence of a unique solution

Question 2

Explain the Uniqueness of an Identity Element

Answer 2

A group has only one identity element, and this can be proven by contradiction:

For a group G under the operation *, let e, f ∈ G, were e and f are identity elements. This means that:
e * f = f, where e is the identity element, and,
e * f = e where f is the identity element.
Since e = e * f, and f = e * f,
=> e = f
Therefore, each group only has one identity element.