Editing Group (mathematics) (section)

=== Symmetry groups ===
{{Main|Symmetry group}}
{{see also|Molecular symmetry|Space group|Point group|Symmetry in physics}}
[[Image:Uniform tiling 73-t2 colored.png|upright=.75|thumb|The (2,3,7) triangle group, a hyperbolic reflection group, acts on this [[Tessellation|tiling]] of the [[hyperbolic geometry|hyperbolic]] plane{{sfn|Ellis|2019}}]]

''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).{{sfn|Weyl|1952}} Conceptually, group theory can be thought of as the study of symmetry.{{efn|More rigorously, every group is the symmetry group of some [[graph (discrete mathematics)|graph]]; see [[Frucht's theorem]], {{harvnb|Frucht|1939}}.}} [[Symmetry in mathematics|Symmetries in mathematics]] greatly simplify the study of [[geometry|geometrical]] or [[Mathematical analysis|analytical]] objects. A group is said to [[Group action (mathematics)|act]] on another mathematical object {{tmath|1= X }} if every group element can be associated to some operation on {{tmath|1= 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 [[Tessellation|tiling]] of the [[hyperbolic plane]] by permuting the triangles.{{sfn|Ellis|2019}} By a group action, the group pattern is connected to the structure of the object being acted on.

In chemistry, [[point group]]s describe [[molecular symmetry|molecular symmetries]], while [[space group]]s describe crystal symmetries in [[crystallography]]. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of [[quantum mechanics|quantum mechanical]] analysis of these properties.<ref>{{harvnb|Conway|Delgado Friedrichs|Huson|Thurston|2001}}. See also {{harvnb|Bishop|1993}}</ref> 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.{{sfn|Weyl|1950|pp=197–202}}

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.{{sfn|Dove|2003}}

Such [[spontaneous symmetry breaking]] has found further application in elementary particle physics, where its occurrence is related to the appearance of [[Goldstone boson]]s.{{sfn|Zee|2010|p=228}}

{| class="wikitable" style="text-align:center; margin:1em auto 1em auto;"
|-
| width=20%|[[Image:C60 Molecule.svg|125px|class=skin-invert-image|alt=A schematic depiction of a Buckminsterfullerene molecule]]
| width=25%|[[Image:Ammonia-3D-balls-A.png|125px|alt=A schematic depiction of an Ammonia molecule]]
| width=15%|[[Image:Cubane-3D-balls.png|125px|alt=A schematic depiction of a cubane molecule]]
| width=20%|[[Image: K2PtCl4.png|125px|class=skin-invert-image]]
|-
| [[Buckminsterfullerene]] displays{{br}}[[icosahedral symmetry]]{{sfn|Chancey|O'Brien|2021|pp=15, 16}}
|[[Ammonia]], NH<sub>3</sub>. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.{{sfn|Simons|2003|loc=§4.2.1}}
|[[Cubane]] C<sub>8</sub>H<sub>8</sub> features{{br}} [[octahedral symmetry]].{{sfn|Eliel|Wilen|Mander|1994|p=82}}
|The [[Potassium tetrachloroplatinate|tetrachloroplatinate(II)]] ion, [PtCl<sub>4</sub>]<sup>2−</sup> exhibits square-planar geometry
|}

Finite symmetry groups such as the [[Mathieu group]]s are used in [[coding theory]], which is in turn applied in [[forward error correction|error correction]] of transmitted data, and in [[CD player]]s.{{sfn|Welsh|1989}} Another application is [[differential Galois theory]], which characterizes functions having [[antiderivative]]s of a prescribed form, giving group-theoretic criteria for when solutions of certain [[differential equation]]s are well-behaved.{{efn|More precisely, the [[monodromy]] action on the vector space of solutions of the differential equations is considered. See {{harvnb|Kuga|1993|pp=105–113}}.}} Geometric properties that remain stable under group actions are investigated in [[geometric invariant theory|(geometric)]] [[invariant theory]].{{sfn|Mumford|Fogarty|Kirwan|1994}}