Group Theory

 

1. Binary Operation:

An operation which is combines two elements of a set to produce another element of the same set is called a binary operation.

Laws of binary operation

Closure law

Let A be a non-empty set on which binary operation '*' is defined, then we say that the set A is closed under the operation '*'.

Thus the set A is closed under the binary operation '*' if a * bÎ A for all aÎA.

Commutative law

Let A be a nonempty set on which a binary operation '*' is defined such that a * b = b * a for all a ÎA, bÎwe say that '*' is commutative over A.

Associative law

Let A be a non-empty set and '*' is a binary operation defined over A such that a * (b * c) = (a * b) * c for all a, b, c Î A, then the binary operation '*' is said to be associative over the set A.

Distributive law

 Let A be a non-empty set and '*', 'o' are two binary operations defined over A such that a * (b o c) = (a * b) o (a * c), then the binary operation * is said to distribute over o in the set.          

Existence law of identity elements: Let A be a non-empty set and a binary operation '*' is defined over A for which the particular element e Î A exists such that a * e = e * a = a for all a Î A.

Then we say that the identity elements exists in a for the binary operation '*'.

Existence law of inverse elements

Let A be a non-empty set and a binary operation '*' defined over A, such that for each a Î A there exists a corresponding elements a¹ Î A such that

a * a¹ = a¹ * a = e where e is the identity element of A for the binary operation '*' over set A, the inverse elements exist and a, a¹ are called inverse of each others.

Group

A non-empty set a, together with a binary operation '*' is called a group (G, *) if the following laws holds for the operation '*'.

Closure law

G is closed under the binary operation '*' i.e. a * b Î G for all a, b Î G.

Associative law

The associative law for the operation '*' holds, i.e. (a * b) * c = a * (b * c) for all a, b, c Î G.

Existence of identity element

The identity element exists in G, i.e. there exists an element e Î G such that

a * e = e * a = a for all a Î G.

Existence if inverse element

The inverse elements of all the elements of G exists, i. e. for a Î G there exists an element b Î G. such that a * b = b * a = e, e being identity element.

Abelian group (Commutative Group)

A Group (G, *) is said to be abelian or commutative group if and only if the operation defined on G is commutative, i. e. iff for all a, b Î G implies that. a * b = b * a