Editing Group (mathematics) (section)

== Examples and applications ==
[[file:Wallpaper group-cm-6.jpg|thumb|right|A periodic wallpaper pattern gives rise to a [[wallpaper group]].]]
Examples and applications of groups abound. A starting point is the group <math>\Z</math> of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains [[multiplicative group]]s. These groups are predecessors of important constructions in [[abstract algebra]].

Groups are also applied in many other mathematical areas. Mathematical objects are often examined by [[functor|associating]] groups to them and studying the properties of the corresponding groups. For example, [[Henri Poincaré]] founded what is now called [[algebraic topology]] by introducing the [[fundamental group]].{{sfn|Hatcher|2002|loc=Chapter I|p=30}} By means of this connection, [[Glossary of topology|topological properties]] such as [[Neighbourhood (mathematics)|proximity]] and [[continuous function|continuity]] translate into properties of groups.{{efn|See the [[Seifert–Van Kampen theorem]] for an example.}}

[[file:Fundamental group.svg|thumb|right|The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers under addition.]]
Elements of the fundamental group of a [[topological space]] are [[equivalence class]]es of loops, where loops are considered equivalent if one can be [[homotopy|smoothly deformed]] into another, and the group operation is "concatenation" (tracing one loop then the other). For example, as shown in the figure, if the topological space is the plane with one point removed, then loops which do not wrap around the missing point (blue) [[null-homotopic|can be smoothly contracted to a single point]] and are the identity element of the fundamental group. A loop which wraps around the missing point <math>k</math> times cannot be deformed into a loop which wraps <math>m</math> times (with {{tmath|1= m\neq k }}), because the loop cannot be smoothly deformed across the hole, so each class of loops is characterized by its [[winding number]] around the missing point. The resulting group is isomorphic to the integers under addition.

In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.{{efn|An example is [[group cohomology]] of a group which equals the [[singular cohomology]] of its [[classifying space]], see {{harvnb|Weibel|1994|loc=§8.2}}.}} In a similar vein, [[geometric group theory]] employs geometric concepts, for example in the study of [[hyperbolic group]]s.{{sfn|Coornaert|Delzant|Papadopoulos|1990}} Further branches crucially applying groups include [[algebraic geometry]] and number theory.<ref>For example, [[class group]]s and [[Picard group]]s; see {{harvnb|Neukirch|1999}}, in particular §§I.12 and I.13</ref>

In addition to the above theoretical applications, many practical applications of groups exist. [[Cryptography]] relies on the combination of the abstract group theory approach together with [[algorithm]]ical knowledge obtained in [[computational group theory]], in particular when implemented for finite groups.{{sfn|Seress|1997}} Applications of group theory are not restricted to mathematics; sciences such as [[physics]], [[chemistry]] and [[computer science]] benefit from the concept.

=== Numbers ===
Many number systems, such as the integers and the [[rational number|rationals]], enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as [[ring (mathematics)|rings]] and fields. Further abstract algebraic concepts such as [[module (mathematics)|module]]s, [[vector space]]s and [[algebra over a field|algebras]] also form groups.

==== Integers ====
The group of integers <math>\Z</math> under addition, denoted {{tmath|1= \left(\Z,+\right) }}, has been described above. The integers, with the operation of multiplication instead of addition, <math>\left(\Z,\cdot\right)</math> do ''not'' form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, <math>a=2</math> is an integer, but the only solution to the equation <math>a\cdot b=1</math> in this case is {{tmath|1= b=\tfrac{1}{2} }}, which is a rational number, but not an integer. Hence not every element of <math>\Z</math> has a (multiplicative) inverse.{{efn|Elements which do have multiplicative inverses are called [[unit (ring theory)|units]], see {{harvnb|Lang|2002|loc=§II.1|p=84}}.}}

==== Rationals ====
The desire for the existence of multiplicative inverses suggests considering [[fraction (mathematics)|fractions]]
<math display=block alt="a/b">\frac{a}{b}.</math>

Fractions of integers (with <math>b</math> nonzero) are known as [[rational number]]s.{{efn|The transition from the integers to the rationals by including fractions is generalized by the [[field of fractions]].}} The set of all such irreducible fractions is commonly denoted {{tmath|1= \Q }}. There is still a minor obstacle for {{tmath|1= \left(\Q,\cdot\right) }}, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no <math>x</math> such that {{tmath|1= x\cdot 0=1 }}), <math>\left(\Q,\cdot\right)</math> is still not a group.

However, the set of all ''nonzero'' rational numbers <math>\Q\smallsetminus\left\{0\right\}=\left\{q\in\Q\mid q\neq 0\right\}</math> does form an abelian group under multiplication, also denoted {{tmath|1= \Q^{\times} }}.{{efn|The same is true for any [[field (mathematics)|field]] <!--use {{math}}, since <math> in footnotes is unreadable on mobile devices-->{{math|''F''}} instead of {{math|'''Q'''}}. See {{harvnb|Lang|2005|loc=§III.1|p=86}}.}} Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of <math>a/b</math> is {{tmath|1= b/a }}, therefore the axiom of the inverse element is satisfied.

The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if [[division (mathematics)|division]] by other than zero is possible, such as in <math>\Q</math> – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.{{efn|For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See {{harvnb|Lang|2002|loc=Theorem IV.1.9}}. The notions of [[Torsion (algebra)|torsion]] of a [[module (mathematics)|module]] and [[simple algebra]]s are other instances of this principle.}}

=== Modular arithmetic ===
{{Main|Modular arithmetic}}

[[File:Clock group.svg|thumb|right|The hours on a clock form a group that uses [[Modular arithmetic|addition modulo]]&nbsp;12. Here, {{nowrap|9 + 4 ≡ 1}}.|alt=The clock hand points to 9 o'clock; 4 hours later it is at 1 o'clock.]]
Modular arithmetic for a ''modulus'' <math>n</math> defines any two elements <math>a</math> and <math>b</math> that differ by a multiple of <math>n</math> to be equivalent, denoted by {{tmath|1= a \equiv b\pmod{n} }}. Every integer is equivalent to one of the integers from <math>0</math> to {{tmath|1= n-1 }}, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent [[representative (mathematics)|representative]]. Modular addition, defined in this way for the integers from <math>0</math> to {{tmath|1= n-1 }}, forms a group, denoted as <math>\mathrm{Z}_n</math> or {{tmath|1= (\Z/n\Z,+) }}, with <math>0</math> as the identity element and <math>n-a</math> as the inverse element of {{tmath|1= a }}.

A familiar example is addition of hours on the face of a [[12-hour clock|clock]], where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on <math>9</math> and is advanced <math>4</math> hours, it ends up on {{tmath|1= 1 }}, as shown in the illustration. This is expressed by saying that <math>9+4</math> is congruent to <math>1</math> "modulo {{tmath|1= 12 }}" or, in symbols,
<math display=block>9+4\equiv 1 \pmod{12}.</math>

For any prime number {{tmath|1= p }}, there is also the [[multiplicative group of integers modulo n|multiplicative group of integers modulo {{tmath|1= p }}]].{{sfn|Lang|2005|loc=Chapter VII}} Its elements can be represented by <math>1</math> to {{tmath|1= p-1 }}. The group operation, multiplication modulo {{tmath|1= p }}, replaces the usual product by its representative, the [[remainder]] of division by {{tmath|1= p }}. For example, for {{tmath|1= p=5 }}, the four group elements can be represented by {{tmath|1= 1,2,3,4 }}. In this group, {{tmath|1= 4\cdot 4\equiv 1\bmod 5 }}, because the usual product <math>16</math> is equivalent to {{tmath|1= 1 }}: when divided by <math>5</math> it yields a remainder of {{tmath|1= 1 }}. The primality of <math>p</math> ensures that the usual product of two representatives is not divisible by {{tmath|1= p }}, and therefore that the modular product is nonzero.{{efn|The stated property is a possible definition of prime numbers. See ''[[Prime element]]''.}} The identity element is represented by&nbsp;{{tmath|1= 1 }}, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer <math>a</math> not divisible by {{tmath|1= p }}, there exists an integer <math>b</math> such that 
<math display=block>a\cdot b\equiv 1\pmod{p},</math>
that is, such that <math>p</math> evenly divides {{tmath|1= a\cdot b-1 }}. The inverse <math>b</math> can be found by using [[Bézout's identity]] and the fact that the [[greatest common divisor]] <math>\gcd(a,p)</math> equals&nbsp;{{tmath|1= 1 }}.{{sfn|Rosen|2000|p=54|loc=&nbsp;(Theorem 2.1)}} In the case <math>p=5</math> above, the inverse of the element represented by <math>4</math> is that represented by {{tmath|1= 4 }}, and the inverse of the element represented by <math>3</math> is represented by&nbsp;{{tmath|1= 2 }}, as {{tmath|1= 3\cdot 2=6\equiv 1\bmod{5} }}. Hence all group axioms are fulfilled. This example is similar to <math>\left(\Q\smallsetminus\left\{0\right\},\cdot\right)</math> above: it consists of exactly those elements in the ring <math>\Z/p\Z</math> that have a multiplicative inverse.{{sfn|Lang|2005|loc=§VIII.1|p=292}} These groups, denoted {{tmath|1= \mathbb F_p^\times }}, are crucial to [[public-key cryptography]].{{efn|For example, the [[Diffie–Hellman]] protocol uses the [[discrete logarithm]]. See {{harvnb|Gollmann|2011|loc=§15.3.2}}.}}

=== Cyclic groups ===
{{Main|Cyclic group}}
[[Image:Cyclic group.svg|right|thumb|upright|The 6th complex roots of unity form a cyclic group. <math>z</math> is a primitive element, but <math>z^2</math> is not, because the odd powers of <math>z</math> are not a power of {{tmath|1= z^2 }}.|alt=A hexagon whose corners are located regularly on a circle]]
A ''cyclic group'' is a group all of whose elements are [[power (mathematics)|powers]] of a particular element {{tmath|1= a }}.{{sfn|Lang|2005|loc=§II.1|p=22}} In multiplicative notation, the elements of the group are
<math display=block>\dots, a^{-3}, a^{-2}, a^{-1}, a^0, a, a^2, a^3, \dots,</math>
where <math>a^2</math> means {{tmath|1= a\cdot a }}, <math>a^{-3}</math> stands for {{tmath|1= a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1} }}, etc.{{efn|The additive notation for elements of a cyclic group would be <!--use {{math}}, since <math> in footnotes is unreadable on mobile devices-->{{math|''t'' ⋅ ''a''}}, where {{math|''t''}} is in {{math|'''Z'''}}.}} Such an element <math>a</math> is called a generator or a [[Primitive root modulo n|primitive element]] of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as
<math display=block>\dots, (-a)+(-a), -a, 0, a, a+a, \dots.</math>

In the groups <math>(\Z/n\Z,+)</math> introduced above, the element <math>1</math> is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are {{tmath|1= 1 }}. Any cyclic group with <math>n</math> elements is isomorphic to this group. A second example for cyclic groups is the group of {{tmath|1= n }}th [[root of unity|complex roots of unity]], given by [[complex number]]s <math>z</math> satisfying {{tmath|1= z^n=1 }}. These numbers can be visualized as the [[vertex (graph theory)|vertices]] on a regular <math>n</math>-gon, as shown in blue in the image for {{tmath|1= n=6 }}. The group operation is multiplication of complex numbers. In the picture, multiplying with <math>z</math> corresponds to a [[clockwise|counter-clockwise]] rotation by 60°.{{sfn|Lang|2005|loc=§II.2|p=26}} From [[field theory (mathematics)|field theory]], the group <math>\mathbb F_p^\times</math> is cyclic for prime <math>p</math>: for example, if {{tmath|1= p=5 }}, <math>3</math> is a generator since {{tmath|1= 3^1=3 }}, {{tmath|1= 3^2=9\equiv 4 }}, {{tmath|1= 3^3\equiv 2 }}, and {{tmath|1= 3^4\equiv 1 }}.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element {{tmath|1= a }}, all the powers of <math>a</math> are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to {{tmath|1= (\Z, +) }}, the group of integers under addition introduced above.{{sfn|Lang|2005|p=22|loc=§II.1 (example 11)}} As these two prototypes are both abelian, so are all cyclic groups.

The study of finitely generated abelian groups is quite mature, including the [[fundamental theorem of finitely generated abelian groups]]; and reflecting this state of affairs, many group-related notions, such as [[Center (group theory)|center]] and [[commutator]], describe the extent to which a given group is not abelian.{{sfn|Lang|2002|loc=§I.5|pp=26, 29}}

=== 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}}

=== General linear group and representation theory ===
{{Main|General linear group|Representation theory|Character theory}}
[[Image:Matrix multiplication.svg|right|thumb|250px|Two [[vector (mathematics)|vectors]] (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the {{tmath|1= x }}-coordinate by factor 2.|alt=Two vectors have the same length and span a 90° angle. Furthermore, they are rotated by 90° degrees, then one vector is stretched to twice its length.]]
[[Matrix group]]s consist of [[Matrix (mathematics)|matrices]] together with [[matrix multiplication]]. The ''general linear group'' <math>\mathrm {GL}(n, \R)</math> consists of all [[invertible matrix|invertible]] {{tmath|1= n }}-by-{{tmath|1= n }} matrices with real entries.{{sfn|Lay|2003}} Its subgroups are referred to as ''matrix groups'' or ''[[linear group]]s''. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the [[special orthogonal group]] {{tmath|1= \mathrm{SO}(n) }}. It describes all possible rotations in <math>n</math> dimensions. [[Rotation matrix|Rotation matrices]] in this group are used in [[computer graphics]].{{sfn|Kuipers|1999}}

''Representation theory'' is both an application of the group concept and important for a deeper understanding of groups.{{sfn|Fulton|Harris|1991}}{{sfn|Serre|1977}} It studies the group by its group actions on other spaces. A broad class of [[group representation]]s are linear representations in which the group acts on a vector space, such as the three-dimensional [[Euclidean space]] {{tmath|1= \R^3 }}. A representation of a group <math>G</math> on an <math>n</math>-[[dimension]]al real vector space is simply a group homomorphism
<math>\rho : G \to \mathrm {GL}(n, \R)</math>
from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.{{efn|This was crucial to the classification of finite simple groups, for example. See {{harvnb|Aschbacher|2004}}.}}

A group action gives further means to study the object being acted on.{{efn|See, for example, [[Schur's Lemma]] for the impact of a group action on [[simple module]]s. A more involved example is the action of an [[absolute Galois group]] on [[étale cohomology]].}} On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and [[topological group]]s, especially (locally) [[compact group]]s.{{sfn|Fulton|Harris|1991}}{{sfn|Rudin|1990}}

=== Galois groups ===
{{Main|Galois group}}
''Galois groups'' were developed to help solve polynomial equations by capturing their symmetry features.{{sfn|Robinson|1996|p=viii}}{{sfn|Artin|1998}} For example, the solutions of the [[quadratic equation]] <math>ax^2+bx+c=0</math> are given by
<math display=block alt="x = (negative b plus or minus the squareroot of (b squared minus 4 a c)) over 2a">x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}.</math>
Each solution can be obtained by replacing the <math>\pm</math> sign by <math>+</math> or {{tmath|1= - }}; analogous formulae are known for [[cubic equation|cubic]] and [[quartic equation]]s, but do ''not'' exist in general for [[quintic equation|degree 5]] and higher.{{sfn|Lang|2002|loc=Chapter VI (see in particular p. 273 for concrete examples)}} In the [[quadratic formula]], changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomial equations and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their [[solvable group|solvability]]) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and [[Nth root|roots]] similar to the formula above.{{sfn|Lang|2002|p=292|loc=(Theorem VI.7.2)}}

Modern [[Galois theory]] generalizes the above type of Galois groups by shifting to field theory and considering [[field extension]]s formed as the [[splitting field]] of a polynomial. This theory establishes—via the [[fundamental theorem of Galois theory]]—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.{{sfn|Stewart|2015|loc=§12.1}}