Gauss developed but would not publish parts from the mathematics of group theory, but Galois is generally considered to be the first to formulate the theory. |

Closure property: a * b $\in$ G, $\forall$ a, b $\in$ G

Associativity: The operation * is associative on G

a * ( b * c) = ( a * b) * c, $\forall$ a, b, c $\in$ G

Existence of Identity:

There exists an element e $\in$ G s.t a * e = a = e * a $\forall$ a $\in$ G

Existence of Inverse:

For each element a $\in$ G, there exists an element b $\in$ G s.t

a * b = e = b * a

The element b is called the inverse of element a with respect to * and we write

b = a$^{-1}$

Some of the very important properties of group are listed below:

Let (G, *) be a group, then

**1)**The identity element e is unique.

**2)**There exists unique inverse in G

i.e., a$^{-1}$ $\in$ G, $\forall$ a $\in$ G.

Also, (a$^{-1}$)$^{-1}$ = a, $\forall$ a $\in$ G and (a * b)$^{-1}$ = b$^{-1} * a^{-1}$, $\forall$ a, b $\in$ G

In general, if a$_{1}, a_{2}, ...................., a_{n}$ are elements of a group G, then

(a$_{1} * a_{2} * ................. * a_{n})^{-1} = a_{n}^{-1} * a_{n-1}^{-1} * ............ * a_{2}^{-1} * a_{1}^{-1}$

**3)**1. a * c = b * c $\rightarrow$ a = b Right cancellation law

2) c * a = c * b $\rightarrow$ a = b ( left cancellation law)

**4)**The left identity is also the right identity.

i.e., ea = a and ae = a

**5)**The left inverse of an element is also its right inverse.

i.e., a$^{-1}$ a = e and aa$^{-1}$ = e

**6)**The equations a * x = b and y * a = b, where, a, b $\in$ G have unique solutions in G, which are x = a$^{-1}$ b $\in$ G and y = b * $^{-1}$ $in$ G respectively. The number of elements in a finite group is called the order of the group. It is denoted by O(G).

An infinite group is called a group of infinite order.

**Examples 1:**The set Z of integers is an infinite abelian group with respect to the operation of addition but Z is not a group with respect to the multiplication.

Let G = {1}, then G is an abelain group of order 1 with respect to multiplication.

Let G = {0}, then G is an abelian group of order 1 with respect to addition.

Let G = {1, -1} then G is an abelian group of order 2 with respect to multiplication.

**Some of the examples based on group theory are solved below for a better understanding of the topic:**

**Example 1:**Show that the set {1 , -1, i, -i} is an abelian finite group of order 4 under multiplication.

**Solution:**

Closure property is satisfied as

1(-1) = - 1, 1. i = i, i(-i) = 1, i(-1) = - i

Associativity: Associative property is satisfied as

(1.i)(-i) = 1.{i(-i)} = 1, {1.i}.(-1) = 1, {i(-1)} = - i etc.,

Existence of Identity:

Axioms on identity is satisfied, 1 being the multiplicative identity,

Existence of inverse:

Axiom on inverse is satisfied, since the inverse on each element of the set exists,

1.1 = e, (-1)(-1) = e = 1, i(-i) = e = 1, (-i)(i) = e = 1

Commutativity:

The commutative law is also satisfied as

1(-1)= (-1).1, (-1)i = i(-1) etc,

Since there are 4 elements in the given set, hence it is a group of order 4.

**Example 2:**For any two elements a and b of a group G, show that G is abelian if and only if(ab)$^{2}$ = a$^{2}$b$^{2}$.

**Solution:**

Let us first suppose that G is abelian

so that ab = ba

$\forall$ a,b $\in$ G

Consider (ab)$^{2}$ = (ab)(ab) = abab associativity

= a(ab)b commutativity

=(aa)(bb) associativity

= a$^{2}$ . b$^{2}$

Thus, (ab)$^{2}$ = a$^{2}$b$^{2}$, $\forall$ a, b $\in$ G

To show ab = ba

consider (ab)$^{2}$ = a$^{2}$b$^{2}$

=> ab(ab) = (aa)(bb)

=> a(ba)b = a(ab)b associativity

ba = ab (by left and right cancellation law)

thus we have ab = ba $\forall$ a,b $in$ G

Hence G is abelian.