Group theory

Group theory is that branch of mathematics concerned with the study of groups.

Elementary introduction

Groups are used throughout mathematics and the sciences, often to capture the internal symmetry of other structures, in the form of automorphism groups. An internal symmetry of a structure is usually associated with an invariant property, and the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group.

Related Topics:
Automorphism group - Invariant - Symmetry group

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In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by the solutions to a polynomial equation. The solvable groups are so-named because of their prominent role in this theory.

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Abelian groups underlie several other structures that are studied in abstract algebra, such as rings, fields, and modules.

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In algebraic topology, groups are used to describe invariants of topological spaces (the name of the torsion subgroup of an infinite group shows the legacy of this field of endeavor). They are called "invariants" because they are defined in such a way that they don't change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups.

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The concept of the Lie group (named for mathematician Sophus Lie) is important in the study of differential equations and manifolds; they combine analysis and group theory and are therefore the proper objects for describing symmetries of analytical structures. Analysis on these and other groups is called harmonic analysis.

Related Topics:
Lie group - Sophus Lie - Differential equations - Manifolds - Harmonic analysis

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In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.

Related Topics:
Combinatorics - Permutation - Burnside's lemma

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An understanding of group theory is also important in the physical sciences. In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. In physics, groups are important because they describe the symmetries which the law of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.

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:Physics examples: Standard Model, Gauge theory

Related Topics:
Standard Model - Gauge theory

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