Editing Group theory (section)

===Chemistry and materials science===
{{Main|Molecular symmetry}}
In [[chemistry]] and [[materials science]], [[point group]]s are used to classify regular polyhedra, and the [[molecular symmetry|symmetries of molecules]], and [[space group]]s to classify [[crystal structure]]s.  The assigned groups can then be used to determine physical properties (such as [[chemical polarity]] and [[Chirality (chemistry)|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 orbital]]s.

[[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.

[[File:Miri2.jpg|thumb|100px|Water molecule with symmetry axis]]
In [[chemistry]], there are five important symmetry operations. They are identity operation ('''E'''), rotation operation or proper rotation ('''C<sub>''n''</sub>'''), reflection operation ('''σ'''), inversion ('''i''') and rotation reflection operation or improper rotation ('''S<sub>''n''</sub>'''). 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 ('''C<sub>''n''</sub>''') 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, {{nowrap|1=''n'' = 2}}, since applying it twice produces the identity operation. In molecules with more than one rotation axis, the C<sub>n</sub> axis having the largest value of n is the highest order rotation axis or principal axis. For example in [[boron trifluoride]] (BF<sub>3</sub>), the highest order of rotation axis is '''C<sub>3</sub>''',  so the principal axis of rotation is '''C<sub>3</sub>'''.

In the reflection operation ('''σ''') 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 '''σ<sub>''h''</sub>''' (horizontal). Other planes, which contain the principal axis of rotation, are labeled  vertical ('''σ<sub>''v''</sub>''') or dihedral ('''σ<sub>''d''</sub>''').

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 [[Tetrahedron|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  ('''S<sub>''n''</sub>''') requires rotation of  360°/''n'', followed by reflection through a plane perpendicular to the axis of rotation.