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Introduction To Group Theory

Group Theory is one of the most important tools in pure mathematics.Not only in mathematics but also in physics and even in our daily life uses of Group Theory is essential.Interesting fact that we use group theory to solve Rubik’s cubes.By using the different permutations and then using the Lagrange’s Theorem the maximum number of moves are calculated to solve the cube.It is also a part of Abstract Algebra.

Group:

An algebraic system(G,) consisting of a non-empty set G together with a composition ,is called a group,if the following condition are satisfied:-
i)## Closure Property:
a∈G, b∈Gab∈G ∀ a,b∈G
ii)## Associate Property:
(ab)c=a(bc) ∀ a,b,c∈G
iii)## Existence of the Identity Element:
There is an element e in G such that
ea=ae=a ∀ a∈G
iv)## Existence of the Inverse of each Element:
Corresponding to each element a∈G there is an element b∈G
Such that
ab=b*a=e
The element b is called the inverse of a.

Abelian Group:

A group (G,) is said to be commutative or abelian ,if the composition is commutative.
i.e. ab=b*a ∀ a,b∈G
i) a+b=b+a ∀ a,b∈G Additive Commutative.
ii)a.b=b.a ∀ a,b∈G Multiplicative Commutative.

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Groupoid:

The algebraic system (G,*) consisting of a non-empty set G and a binary composition * defined on G is known as Groupoid.

Semi-group:

A groupoid(G,*) is called a Semi-group,if the composition on G is associative.

Example:

i)The set of natural number is a semi-group under addition.
ii)Set of irrational numbers under addition and multiplication is semi-group.

Monoid:

A semi-group(G,*) in which the composition admits an identity elementin G,is known as monoid.If any monoid satisfy inverse law then it is called a group.
Relation is not a part of set.

Example:

i)Set of natural numbers under multiplication is a monoid.

Ring:

An algebraic structure(R,+,.) consisting of a non-empty set R with two binary compositions under addition and multiplication,is called a ring,if the following axioms are satisfied.
i)(R,+) is an abelian group.
ii)under multiplication semi-group or associative.i.e (ab)c=a(bc)
∀ a,b,c∈G.
iii)Distributive law holds.i.e.
a(b+c)=ab=ac and (ab)c=ac+bc ∀ a,b,c∈R.

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Homomorphism of a Group:

Let(G,0) and (G’,) be two groups.then a mapping f from G to G’ is called a homomorphism,if f is composition preserving.i.e
F(a0b) = f(a)f(b) ∀ a,b∈G.

Example:

The additive group of all integers is homomrphism.

Isomorphism of a Group:

Let (G,0) and (G’,) be two groups.Then a one-one onto composition preserving mapping from G to G’ is called an isomorphism.In this case,we say that (G,0) is isomorphic to (G’,) and we write
(G,0)≅(G,*).

Automorphism:

An isomorphism of a group ,onto itself is called an automorphism.
Example:
F(x)=-x ∀ x∈I is an automorphism for the mapping f:I->I.

Tricks To Remember:

i)Homomorphism one-one ,onto ->Isomorphism.
ii)Isomorphism onto itself ->Automorphism
iii)Homomorphism one-one ->Monomorphism

Reference Book:Modern Algebra

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