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Group Theory

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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

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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

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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

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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

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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.