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R group may refer to: In chemistry: Pendant group or side group; Side chain; Substituent; In mathematics: Tempered representation This page was last edited on ...
[r] A group action gives further means to study the object being acted on. [s] On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups. [63] [65]
For example, the dihedral group D 8 of order sixteen can be generated by a rotation, r, of order 8; and a flip, f, of order 2; and certainly any element of D 8 is a product of r ' s and f ' s. However, we have, for example, rfr = f −1, r 7 = r −1, etc., so such products are not unique in D 8. Each such product equivalence can be expressed ...
If R is a ring or an algebra over a field, and S is a subset of R, then the centralizer of S is exactly as defined for groups, with R in the place of G.. If is a Lie algebra (or Lie ring) with Lie product [x, y], then the centralizer of a subset S of is defined to be [4]
Such a ring homomorphism R → End Z (M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it. Such a representation R → End Z (M) may also be called a ring action of R on M.
If G is a group and R is a ring, the group ring of G over R is a free module over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G. The ring of differential operators (depending on the context). In fact, many rings that appear in ...
More generally, the fundamental group of a bouquet of r circles is the free group on r letters. The fundamental group of a wedge sum of two path connected spaces X and Y can be computed as the free product of the individual fundamental groups: () ().
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.