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R groupmay refer to: In chemistry: Pendant groupor side group. Side chain. Substituent. In mathematics: Tempered representation. Topics referred to by the same term. This disambiguationpage lists articles associated with the title R group.
The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if rn = 1, then rn−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R× is not closed under addition. A nonzero ring R in which every nonzero element is a unit (that is, R ...
Formally, a group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the underlying set of the group, and the operation is called the group operation or the group law. A group and its underlying set are thus two different mathematical objects.
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
Define a map from R × to the set of all nonzero fractional ideals of R by sending every element to the principal (fractional) ideal it generates. This is a group homomorphism; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map ...
General linear group. 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 ...
In the of the elements, each column is a group. In chemistry, a group (also known as a family) [1] is a column of elements in the periodic table of the chemical elements. There are 18 numbered groups in the periodic table; the 14 f-block columns, between groups 2 and 3, are not numbered. The elements in a group have similar physical or chemical ...
Carboxylate. In organic chemistry, a carboxylate is the conjugate base of a carboxylic acid, RCOO− (or RCO− 2). It is an anion, an ion with negative charge. Carboxylate salts are salts that have the general formula M (RCOO)n, where M is a metal and n is 1, 2,.... Carboxylate esters have the general formula RCOOR′ (also written as RCO2R ...