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# Group Theory

Top
 Sub Topics 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. Group theory is usually a powerful formal opportunity for analyzing abstract as well as physical systems where symmetry is present and it has surprising importance inside physics, especially quantum motion. Group is a few elements together through an operation that mixes any two associated with its elements to a third factor satisfying four circumstances called the group axioms, namely closure, associativity, identity as well as invertibility. The concept of a group arose from the study of polynomial equations. The axioms for a group are short and natural.

## Definition

Let G be a non empty set and * be a binary operation defined on it, then the structure (G, *) is said to be a group, if the following axioms are satisfied,

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$^{-1}$

## Properties of Groups

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.

## Order of a group

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.

## Examples

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.