# Group Theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced tremendous advances and have become subject areas in their own right.

Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many applications in physics and chemistry.

One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.

### Other articles related to "group theory, theory, group, groups":

History Of Group Theory - Early 19th Century - Convergence
... Group theory as an increasingly independent subject was popularized by Serret, who devoted section IV of his algebra to the theory by Camille Jordan, whose Traité des substitutions et des ... Other group theorists of the 19th century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Émile Mathieu as well as Burnside, Dickson, Hölder, Moore, Sylow, and Weber ... The convergence of the above three sources into a uniform theory started with Jordan's Traité and von Dyck (1882) who first defined a group in the ...
Combinatorics On Words - Group Theory - Considered Problems
... One problem considered in the study of combinatorics on words in group theory is the following for two elements x,y of a semigroup, does x=y modulo the defining relations of x and y ... Undecidable means the theory cannot be proved ... This question asks if a group is finite if the group has a definite number of generators and meets the criteria xn=1, for x in the group ...
Applications of Group Theory
... Applications of group theory abound ... Almost all structures in abstract algebra are special cases of groups ... Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication) ...
List Of University Of Manchester People - Natural and Applied Sciences - Mathematics
... Frank Adams was a leading figure in algebraic topology and homotopy theory ... He developed methods which led to important advances in calculating the homotopy groups of spheres (a problem which is still unsolved), including the invention of the Adams ... He is also known for his work in the theory of statistical inference and in multivariate analysis ...
History Of Group Theory - Late 20th Century
... century enjoyed the successes of over one hundred years of study in group theory ... In finite groups, post classification results included the O'Nan–Scott theorem, the Aschbacher classification, the classification of multiply transitive finite groups, the determination of the ... The modular representation theory entered a new era as the techniques of the classification were axiomatized, including fusion systems, Puig's theory of pairs and nilpotent blocks ...

### Famous quotes containing the words theory and/or group:

Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.
Willard Van Orman Quine (b. 1908)

No group and no government can properly prescribe precisely what should constitute the body of knowledge with which true education is concerned.
Franklin D. Roosevelt (1882–1945)