Symmetry group

 

Symmetry groups[edit]

Main article: Symmetry group
See also: Molecular symmetry, Space group, Point group, and Symmetry in physics
The (2,3,7) triangle group, a hyperbolic reflection group, acts on this tiling of the hyperbolic plane[50]

Symmetry groups are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below).[51] Conceptually, group theory can be thought of as the study of symmetry.[p] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element can be associated to some operation on X and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles.[50] By a group action, the group pattern is connected to the structure of the object being acted on.

In chemistry, point groups describe molecular symmetries, while space groups describe crystal symmetries in crystallography. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties.[52] For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.[53]

Group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.[54]

Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.[55]

A schematic depiction of a Buckminsterfullerene moleculeA schematic depiction of an Ammonia moleculeA schematic depiction of a cubane molecule
Buckminsterfullerene displays
icosahedral symmetry[56]		Ammonia, NH3. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.[57]Cubane C8H8 features
octahedral symmetry.[58]		The tetrachloroplatinate(II) ion, [PtCl4]2- exhibits square-planar geometry

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players.[59] Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.[q] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.[60]

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