## 1.2 Properties of Groups

Recall the definition of a group:

A set G is “upgraded” into a **group** if it satisfied the following axioms under **one** binary operation (*) :

- Closure:
- Associativity:
- Identity:
- Inverse:

An **Abelian group** is a group that is follows the axioms 1 – 4 with the addition of one property:** **

- Commutativity:

In addition to the axioms, the following properties of groups are important to note:

- Uniqueness of the identity element
- Uniqueness of the inverse element
- Cancellation law
- Inverse property (extended)

Uniqueness of an element in mathematics means there exists only one such element with that property. We prove uniqueness by making an assumption that there are two elements in the set that satisfy the property, and show that if such a situation holds, then the two elements must be equal!

We use * to denote the binary operation between elements and “QED” to signal the end of the proof.

The remainder of the post aims to go through the proofs of these properties!…