In the theory of Lie groups in mathematics, especially those that are compact, a special role is played by the torus groups. In a compact Lie group G there is to be found a maximal torus T; that is, a closed subgroup that is a torus, and of the largest possible dimension. That dimension is called the rank of G. The rank occurs as the number of nodes in the Dynkin diagram of a semisimple group. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... // Geometry In geometry, a torus (pl. ... In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation... See also Simple Lie group. ...

For example, the Lie group SO(3) of rotations in three dimensions has as maximal torus T a circle group (a 1-torus, that is). It can be taken to be the group of rotations about the x-axis, parametrised by angle. According to general theory, all the maximal tori form a single conjugacy class of subgroups. The related group SU(2) also has rank 1, with a rotation group as maximal torus. The conjugacy of maximal tori implies that all the maximal tori SO(3) are the rotations about some fixed axis - so that we have surveyed them all. In general SO(2n) and SO(2n+1) have rank n. In those cases one can easily find explicit parameter angles for the maximal torus: that is, commuting one-parameter families of rotations exhibiting the torus as a product of circle groups. In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...

The Weyl group W of G is the normalizer of T in G modulo the centralizer; or in other words the group of transformations of T into itself carried out by conjugation in G. The representation theory of G, when it is a connected group at least, is essentially determined by T and W. Qwertyuiop. In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...