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It is called the general linear group, and denoted GL n (R) or GL(n, R) (where R is the set of real numbers). Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n -dimensional Euclidean space that fix a given point (the origin).
For example, R-22 has one carbon atom, one hydrogen atom (2−1 = 1), two fluorine atoms, and one chlorine atom (4−2−1 = 1), so it is chlorodifluoromethane, while R-134 has two carbon atoms (2−1 = 1), two hydrogen atoms (3−1 = 2), four fluorine atoms, and no chlorine atoms (6−2−4 = 0), so it is one of the tetrafluoroethanes. This ...
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 ...
Basic examples are orthogonal, unitary and symplectic groups but it is possible to construct more using division algebras (for example the unit group of a quaternion algebra is a classical group). Note that the projective groups associated to these groups are also linear, though less obviously.
This holds in particular for any representation of a finite group over the complex numbers, since the characteristic of the complex numbers is zero, which never divides the size of a group. In the example above, the first two representations given (ρ and σ) are both decomposable into two 1-dimensional subrepresentations (given by span{(1,0 ...
SL(2, R) is the group of all linear transformations of R 2 that preserve oriented area. It is isomorphic to the symplectic group Sp(2, R) and the special unitary group SU(1, 1). It is also isomorphic to the group of unit-length coquaternions. The group SL ± (2, R) preserves unoriented area: it may reverse orientation.
The general formula of a carboxylic acid is often written as R−COOH or R−CO 2 H, sometimes as R−C(O)OH with R referring to an organyl group (e.g., alkyl, alkenyl, aryl), or hydrogen, or other groups. Carboxylic acids occur widely. Important examples include the amino acids and fatty acids. Deprotonation of a carboxylic acid gives a ...
The simplest example is tert-butanol (2-methylpropan-2-ol), for which each of R, R', and R" is CH 3. In these shorthands, R, R', and R" represent substituents , alkyl or other attached, generally organic groups.