Dear Rahul,

Group and ring theory are central in many ways. Conservation laws of physics are reflections of the principle of least action. Once you have one of these laws in place, then your immediate concern is what actions can you take which preserve the law, and that set of actions is generally a group. This was a big understanding arrived at by Emmy Noether.

Ring theory has many uses as well. As Rama Bandi mentioned above, it is useful in coding theory, and number theory in general, e.g., cryptography.

Semigroups are to do with actions that preserve partial symmetries, and can be used to model plant growth or the growth of quasi-crystals, and etc..

Basically, these algebraic structures are useful for understanding how one can transform a situation given various degrees of freedom, and as this is a fundamental type of question, these structures end up being essential.

Projective geometry and lots of other fields.

The theory behind permutation groups has a number of applications in solving graph isomorphism-related problems, which have many applications in the study of network topology and even data bases.

Most of the coding theorists are working on rings.................Dus to the structure of the rings 1 can find some good codes over rings than finite fields...........

Similarly in cryptography, many researchres r usings rings................

Dear Rahul,

Group and ring theory are central in many ways. Conservation laws of physics are reflections of the principle of least action. Once you have one of these laws in place, then your immediate concern is what actions can you take which preserve the law, and that set of actions is generally a group. This was a big understanding arrived at by Emmy Noether.

Ring theory has many uses as well. As Rama Bandi mentioned above, it is useful in coding theory, and number theory in general, e.g., cryptography.

Semigroups are to do with actions that preserve partial symmetries, and can be used to model plant growth or the growth of quasi-crystals, and etc..

Basically, these algebraic structures are useful for understanding how one can transform a situation given various degrees of freedom, and as this is a fundamental type of question, these structures end up being essential.

Not only ring theory, group theory is also used in cryptography. MOR crypto system uses non-abelian groups and their automorphisms. Even the crypto systems used for practical purposes use group theory along with number theory and other things.

Groups and rings are used everywhere. This is because the former crop up whenever there is some symmetry (not the least coming from redundancy in description) whereas the latter crops up whenever some sort of  arithmetic crops up (i.e. there is some addition, substraction and multiplication defined, where the usual associativity and distributivity rules hold, but division may not be possible) 

Example: groups:

*The matrix group O(3), that is the group of all linear transformations preserving the innerproduct (i.e. a "symmetry")  on euclidean 3 space.

*The symmetric group S(7): the group of enumerations of seven things.

*The Galois group of the equation X^3 + X^2 + X + 1 = 0 , the group  of field automorphisms of Q(i) over the rationals. It permutes all primitive 4th roots of unity (i.e. i an -i)

Example  Rings.

*The integers.

*The ring C([0,1]) of continuous functions on [0,1]

*The ring of polynomials C[X,Y]

*The ring of polynomials defined on the elliptic curve Y^2 = X^3 + X + 1,

C[X, Y}/(Y^2  - X^3 - X - 1),  

where (Y^2 - X^3 - X -1) is the _ideal_ generated by Y^2 - X^3 - X - 1

* The ring of n*n matrices on R^n.

* the quaternions.

* the ring of bounded (i.e. continuous) linear operators on a Hibert space. 

The latter three are examples of non commutative rings, whereas the first is an example of a ring that does not contain a field (rings that contain a field are often called algebra's). 

Almost everywhere in physics. In mechanics, in crystalography

http://en.wikipedia.org/wiki/Crystallographic_point_group

in particle physics

http://en.wikipedia.org/wiki/Particle_physics_and_representation_theory

Wherever there exist a symmetry of some kind. 

http://en.wikipedia.org/wiki/Symmetry_%28physics%29

https://www.youtube.com/watch?v=3wNPrSwbtQ8

Why is group theory important?

Broadly speaking, group theory is the study of symmetry. When we are dealing with an object that appears symmetric, group theory can help with the analysis. We apply the label symmetric to anything which stays invariant under some transformations. This could apply to geometric figures (a circle is highly symmetric, being invariant under any rotation), but also to more abstract objects like functions: x2 + y2 + z2 is invariant under any rearrangement of x, y, and z and the trigonometric functions sin(t) and cos(t) are invariant when we replace t with t+2π.

Modern particle physics would not exist without group theory; in fact, group theory predicted the existence of many elementary particles before they were found experimentally.

Details at http://www.math.uconn.edu/~kconrad/math216/whygroups.html

any specific real example which can be solved using group, ring and field?