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!

uniqueness of the identity element proof

We want to show that the identity element of a group is unique. Let’s assume the opposite.

**Proof:**

Suppose for group G, there are two identity elements, and Then for any element,

So This shows that (hence if there are two identity elements in a set, then those elements are the same thing just with different labelling)!

**QED**

To break this proof, look at the following bullet points:

- we can say because here acts as the identity element, and acts as a regular element of the set
- holds since identity elements are commutative, by definition

(RECALL) Identity: . This shows the commutative nature of identity elements

- finally we conclude since we use the fact that can also act as a regular element and can act as the identity element

uniqueness of inverse element proof

We want to show that the inverse element is unique. Suppose the opposite

**Proof:**

Suppose there exists two inverse elements, Then where is the identity element of G.

Then Hence,

**QED**

Again, I’ll breakdown the proof below:

- using the identity property, we can say
- since we defined the inverse of as: then we can substitute for
- by associativity of G,
- and by assumption we can substitute for
- and finally, we use the identity property again to conclude

cancellation laws proof

In a group G, if then for

**Proof:**

Suppose

Let h be the inverse of g, then for identity element,

Then as required

**QED**

The breakdown is as follows:

- is possible using the identity property
- is possible since hence used substitution
- by associativity
- since we assumed
- by associativity
- by inverse property
- by identity property

inverse of

**Proof:**

Let then we know satisfying and

We claim is the inverse of . To show this, we want to show that

Since

And

Then

**QED**

The breakdown:

- by associativity
- by inverse property
- by identity property
- by identity property
- since both equal

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