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Fucked by a Shemale 2 2011 Fuckedbyshemale.com DO Ericka, Carol and Kawanna 2009 grooby.com MastOnly Dominating Shemales Punish This Slave 2009 TrannyPack.com Top Dominant Shemales Owning Their Slave 2009 TrannyPack.com Top Dark Skinned Shemale Cock 2010 Universal Trannies Top Bottom DO Cum Swapping Tranny 2008 Zaye Entertainment CumSwap Bottom DR Big Dick Shemale Beauties 1 2010 Universal Trannies Top Bottom unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem. Harmonic analysis Main article: Harmonic analysis Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques.[12] Combinatorics 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. The circle of fifths may be endowed with a cyclic group structure Music The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. Transformational theory models musical transformations as elements of a mathematical group.

Physics In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group. Chemistry and materials science In chemistry and materials science, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as chemical polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct molecular orbitals. Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule. Water molecule with symmetry axis In chemistry, there are five important symmetry operations. They are identity operation (E), rotation operation or proper rotation (Cn), reflection operation (s), inversion (i) and rotation reflection operation or improper rotation (Sn). The identity operation (E) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis (Cn) consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/n, where n is an integer, about a rotation axis. For example, if a water molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen atoms, it is in the same configuration as it started. In this case, n = 2, since applying it twice produces the identity operation. In molecules with more than one rotation axis, the Cn axis having the largest value of n is the highest order rotation axis or principal axis. For example Borane (BH3), the highest order of rotation axis is C3, so Principal axis of rotation of axis is C3. In the reflection operation (s) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called sh (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (sv) or dihedral (sd). Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started.Many molecules that seem at first glance to have an inversion center do not; for example, methane and other tetrahedral molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation (Sn) requires rotation of 360°/n , followed by reflection through a plane perpendicular to the axis of rotation. Statistical mechanics Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Willard Gibbs, relating to the summing of an infinite number of probabilities to yield a meaningful solution.[13] Cryptography Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group. The cyclic group Z26 underlies Caesar's cipher. History Main article: History of group theory 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. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

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