A group in mathematics is a set G along with a binary operation (henceforth denoted as \star ) satisfying four axioms.

By a binary operation, we mean an operation that takes two elements of a group and returns back another element of the group. For example, “addition” on the set of integers is an example of a binary operation because when we add two integers we get back another integer.

**The Four Group Axioms**:

We say that a set G along with a binary operation \star is a group if it satisfies the following four axioms:

### 1. Closure

Given any two elements a \text{ and } b \text{ in } G we have that a\star b\in G .

This means that the set G is closed under the operation \star . This is precisely what we mean by saying that \star is a binary operation on G.

### 2. Associativity

Given any any three elements a, b \text{ and } c \text{ in } G we have that:

(a\star b)\star c = a\star(b\star c) This means that we can “associate” the three group elements in any two ways we desire.

### 3. Existence of Identity

There exists an element e \in G such that :

a\star e = a = e\star a for all a \in G

The element e is called the identity element because it leaves any element of the group unchanged when operated with it.

### 4. Existence of Inverses

For any element a \in G there exists an element b \text{ in } G such that :

a\star b = e = b\star aThe element b is called the inverse of a and is denoted by the symbol a^{-1}

## Abelian/Commutative Group:

In addition to the above four group axioms if the group also satisfies the axiom of commutativity then we say that the group is abelian. The axiom of commutativity says that,

Given any two elements a \text{ and } b \text{ in } G we have that:

a\star b = b\star a**Examples of Groups**:

Let us look at some examples of groups and verify the group axioms for them:

#### 1. Integers with addition

The set of integers (denoted by Z) along with the usual addition operation as our binary operation forms a group. We can verify that it satisfies the four axioms:

- Closure – The sum of two integers is once again clearly an integer. Therefore the axiom of closure is satisfied.
- Associativity – When adding three integers we can add them is any of the two ways. For example, (2+3)+5=2+(3+5)
- Existence of Identity – The number 0 serves as the identity element for the set of integers. We have that, a + 0 = a for all integers a.
- Existence of Inverses – For any integer a, the integer (-a) serves as the additive inverse because a+(-a) = 0.

In fact the addition operation is commutative for the set of integers so we see that the (Z,+) is an abelian group.

#### 2. Nonzero rational numbers with multiplication

The set of non-zero rationals (denoted by Q*) along with the usual multiplication operation as our binary operation forms a group. We can verify that it satisfies the four axioms:

- Closure – The product of two rationals is once again clearly a rational numbers. Therefore the axiom of closure is satisfied.
- Associativity – When multiplying three rationals we can multiply them is any of the two ways. For example, (3X4)x5=3x(4×6)
- Existence of Identity – The number 1 serves as the identity element for the set of non zero rationals. We have that, a x 1 = a for any nonzero rational numbers a.
- Existence of Inverses – For any rational a, the rational number (1/a) serves as the multiplicative inverse because ax(1/a) = 1. Notice that it was necessary to remove 0 from the set of rationals since 0 does not have a multiplicative inverse.

Moreover the multiplication operation is commutative for the set of nonzero rationals and so we see that the (Q*,+) is an abelian group.