Editing Group theory (section)

==Applications of group theory==<!--this section is linked at from [[Applications of group theory]]-->
Applications of group theory abound. Almost all structures in [[abstract algebra]] are special cases of groups. [[Ring (mathematics)|Rings]], for example, can be viewed as [[abelian group]]s (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.

===Galois theory===
{{Main|Galois theory}}
[[Galois theory]] uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The [[fundamental theorem of Galois theory]] provides a link between [[algebraic field extension]]s and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding [[Galois group]]. For example, ''S''<sub>5</sub>, the [[symmetric group]] in 5 elements, is not solvable which implies that the general [[quintic equation]] cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as [[class field theory]].

===Algebraic topology===
{{Main|Algebraic topology}}
[[Algebraic topology]] is another domain which prominently [[functor|associates]] groups to the objects the theory is interested in. There, groups are used to describe certain invariants of [[topological space]]s. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some [[homeomorphism|deformation]]. For example, the [[fundamental group]] "counts" how many paths in the space are essentially different. The [[Poincaré conjecture]], proved in 2002/2003 by [[Grigori Perelman]], is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of [[Eilenberg–MacLane space]]s which are spaces with prescribed [[homotopy groups]]. Similarly [[algebraic K-theory]] relies in a way on [[classifying space]]s of groups. Finally, the name of the [[torsion subgroup]] of an infinite group shows the legacy of topology in group theory.

[[File:Torus.png|thumb|right|200px|A torus. Its abelian group structure is induced from the map {{nowrap|'''C''' → '''C'''/('''Z''' + ''τ'''''Z''')}}, where ''τ'' is a parameter living in the [[upper half plane]].]]

===Algebraic geometry===
{{Main|Algebraic geometry}}
[[Algebraic geometry]] likewise uses group theory in many ways. [[Abelian variety|Abelian varieties]] have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. (For example the [[Hodge conjecture]] (in certain cases).) The one-dimensional case, namely [[elliptic curve]]s is studied in particular detail. They are both theoretically and practically intriguing.<ref>See the [[Birch and Swinnerton-Dyer conjecture]], one of the [[millennium problem]]s</ref> In another direction, [[toric variety|toric varieties]] are [[algebraic variety|algebraic varieties]] acted on by a [[torus]]. Toroidal embeddings have recently led to advances in [[algebraic geometry]], in particular [[resolution of singularities]].<ref>{{Citation | last1=Abramovich | first1=Dan | last2=Karu | first2=Kalle | last3=Matsuki | first3=Kenji | last4=Wlodarczyk | first4=Jaroslaw | title=Torification and factorization of birational maps | mr=1896232  | year=2002 | journal=[[Journal of the American Mathematical Society]] | volume=15 | issue=3 | pages=531–572 | doi=10.1090/S0894-0347-02-00396-X| arxiv=math/9904135 | s2cid=18211120 }}</ref>

===Algebraic number theory===
{{Main|Algebraic number theory}}
[[Algebraic number theory]] makes uses of groups for some important applications. For example, [[Euler product|Euler's product formula]],
:<math>
\begin{align}
\sum_{n\geq 1}\frac{1}{n^s}& = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}, \\
\end{align}
\!</math>
captures [[Fundamental theorem of arithmetic|the fact]] that any integer decomposes in a unique way into [[prime number|primes]]. The failure of this statement for [[Dedekind ring|more general rings]] gives rise to [[class group]]s and [[regular prime]]s, which feature in [[Ernst Kummer|Kummer's]] treatment of [[Fermat's Last Theorem]].

===Harmonic analysis===
{{Main|Harmonic analysis}}
Analysis on Lie groups and certain other groups is called [[harmonic analysis]]. [[Haar measure]]s, that is, integrals invariant under the translation in a Lie group, are used for [[pattern recognition]] and other [[image processing]] techniques.<ref>{{Citation | last1=Lenz | first1=Reiner | title=Group theoretical methods in image processing | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Computer Science | isbn=978-0-387-52290-6 | year=1990 | volume=413 | url=https://archive.org/details/grouptheoretical0000lenz | doi=10.1007/3-540-52290-5 | s2cid=2738874 | url-access=registration }}</ref>

===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]].

[[File:Fifths.png|right|thumb|150px|The circle of fifths may be endowed with a cyclic group structure.]]

===Music===
The presence of the 12-[[Periodic group|periodicity]] in the [[circle of fifths]] yields applications of [[elementary group theory]] in [[set theory (music)|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 (physics)|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]].

Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by [[Josiah Willard Gibbs|Willard Gibbs]], relating to the summing of an infinite number of probabilities to yield a meaningful solution.<ref>[[Norbert Wiener]], Cybernetics: Or Control and Communication in the Animal and the Machine,  {{ISBN|978-0262730099}}, Ch 2</ref>

===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.

===Cryptography===

[[File:Caesar3.svg|thumb|The [[cyclic group]] '''Z'''<sub>26</sub> underlies [[Caesar's cipher]].]]
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 cipher|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 group]]s. So the term [[group-based cryptography]] refers mostly to [[cryptographic protocol]]s that use infinite [[non-abelian group]]s such as a [[braid group]].