Group Theory

It is a segment of Modern Algebra which deals with the study of groups. Groups refer to those systems which consist of a set of elements and a binary operation that can be applied to certain two elements contained in a set, which together satisfy certain properties or axioms.

In order for a system to be considered as a group, it must satisfy the following axioms or properties.

Closure Property

Require that the group be closed under the operation. The combination of any two elements which both belong to the same set of numbers produces another element of the same set when subjected to a certain binary operation.

Example:

Consider the system (R, + ):

let a = 5, b = - 7

both a and b belong to the set of Real numbers.

then 5 + - 7 = - 2 , -2 belongs to R.

Associative Law

let us check if the given system obeys Associative Law:

let c = 10

( a + b ) + c = a + ( b + c )

(5 + -7) + 10 = 5 + ( -7 + 10)

-2 + 10 = 5 + 3

8 = 8

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Presence of an identity element which also belong to the same set R.

An identity element is a member of the same set and when added to any element of the set, such element retains its identity.

Let i = 0

5 + 0 = 5

0 also belong to the set of Real Numbers.

Presence of an Inverse

The inverse of a number when added to the original number must yield a value equal to the identity element.

Let a = 5 Inverse = - 5

5 + - 5 = 0

-5 also belong to the set of Real Number.

0 is the identity element of { R, + }.

Thus the system { R, + } is a group since it satisfies the Closure, Associative, Identity and Inverse laws.

{ Q, * } where Q is a set of Rational numbers and * refers to multiplication is not a group since 0 has no multiplicative inverse (1/0 is undefined).

Abelian Group

A group does not necessarily satisfy commutative law, but a group that satisfies commutative law belongs to the group known as "Abelian Group." Thus the Abelian group satisfies axioms on Closure, Associative, Identity, Inverse and Commutative.

Source:

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