Group Theory - E-book - ePub

By many expert mathematicians, group theory is oftenaddressed as a central part of mathematics. It finds its originsin geometry, since geometry describes groups in a detailedmanner. The theory of polynomial equations also describes theprocedure and principals of associating a finite group with anypolynomial equation. This association is done in such a waythat makes the group to encode information that can be usedto solve the equations.
This equation theory was developed byGalois. Finite group theory faced a number of changes in nearpast times as a result of classification of finite simple groups. The most important theorem when practicing group theory istheorem by Jordan holder. This theorem shows how any finitegroup is a combination of multiple finite simple groups. Group theory is a term that is mainly used fields related tomathematics such as algebraic calculations.
In abstract algebra, groups are referred as algebraic structures. Other terms ofalgebraic theories, such as rings, fields and vector spaces arealso seen as group. Of course with some additional operationsand axioms, mathematicians accept them as a group. Themethods and procedures of group theory effect many partsand concepts of mathematics as well as algebra on a largescale. Linear algebraic groups and lie groups are two mainbranches or say categories of group theory that have advancedenough to be considered as a subject in their ownperspectives.
Not only mathematics, group theory also finds its roots invarious physical systems, especially in crystals and hydrogenatom. They might be modeled by symmetry groups. Thus itcan be said that group theory possess close relations withrepresentation theory. Principals and ideas of group theory arepractically applied in the fields of physic, material science andchemistry of course. Group theory is also considered as acentral key in the studies and practices of cryptography.
In2000s, more than 10000 pages were published in the timespan of 1960 to 1980. These publications were a collaborativeeffort in order to culminating the result as a completeclassification of infinite simple groups. For the practitioners andlearners of mathematics or even physics the theory of groupshas a great importance. Not all aspects of this theory are usedin mathematics or physics. But there are some ideas andprincipals that help a lot as you advance to higher levelmathematics, it is very same with the physics.
Full application ofthis wide theory is not possible on a single subject anyhow. However it is partially applied in both cases, and still leaves agreat influence.