This in an excerpt from this book
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.
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