By many expert mathematicians, group theory is often

addressed as a central part of mathematics. It finds its origins

in geometry, since geometry describes groups in a detailed

manner. The theory of polynomial equations also describes the

procedure and principals of associating a finite group with any

polynomial equation. This association is done in such a way

that makes the group to encode information that can be used

to solve the equations. This equation theory was developed by

Galois. Finite group theory faced a number of changes in near

past times as a result of classification of finite simple groups.

The most important theorem when practicing group theory is

theorem by Jordan holder. This theorem shows how any finite

group is a combination of multiple finite simple groups.

Group theory is a term that is mainly used fields related to

mathematics such as algebraic calculations. In abstract algebra,

groups are referred as algebraic structures. Other terms of

algebraic theories, such as rings, fields and vector spaces are

also seen as group. Of course with some additional operations

and axioms, mathematicians accept them as a group. The

methods and procedures of group theory effect many parts

and concepts of mathematics as well as algebra on a large

scale. Linear algebraic groups and lie groups are two main

branches or say categories of group theory that have advanced

enough to be considered as a subject in their own

perspectives.

Not only mathematics, group theory also finds its roots in

various physical systems, especially in crystals and hydrogen

atom. They might be modeled by symmetry groups. Thus it

can be said that group theory possess close relations with

representation theory. Principals and ideas of group theory are

practically applied in the fields of physic, material science and

chemistry of course. Group theory is also considered as a

central key in the studies and practices of cryptography. In

2000s, more than 10000 pages were published in the time

span of 1960 to 1980. These publications were a collaborative

effort in order to culminating the result as a complete

classification of infinite simple groups. For the practitioners and

learners of mathematics or even physics the theory of groups

has a great importance. Not all aspects of this theory are used

in mathematics or physics. But there are some ideas and

principals that help a lot as you advance to higher level

mathematics, it is very same with the physics. Full application of

this wide theory is not possible on a single subject anyhow.

However it is partially applied in both cases, and still leaves a

great influence.