Group (mathematics) 19447 226048996 2008-07-16T16:29:59Z RobHar 1520818 /* General references */ replace fulton-harris ref with template ref {{hatnote|This article covers only the basic notions related to groups. Further ways of studying groups are treated in [[group theory]].}} [[Image:Rubik's cube.svg|thumb|right|200px|The possible rearrangements of [[Rubik's Cube]] form a group, called the [[Rubik's Cube group]].]]{{Groups}} In [[mathematics]], a '''group''' is a fundamental object of the field of [[abstract algebra]]. A group is a [[set (mathematics)|set]] of [[element (mathematics)|elements]] with a single operation by which two elements may be combined into a third; to qualify as a group, the operation must satisfy several conditions. Groups play an important role in many fields, and form the core of several more complex algebraic objects such as [[ring (mathematics)|rings]] and [[module (mathematics)|modules]].<ref>{{Harvard citations|last = Herstein|year = 1975|loc = &#xa7;2, p. 26|nb = yes}}</ref><ref>{{Harvard citations|last = Hall|year = 1967|loc = &#xa7;1.1, p. 1|nb = yes}}: "The idea of a group is one which pervades the whole of mathematics both pure and applied."</ref> Since the definition of a group is so general, groups occur in many forms and have applications in many areas in- and outside of mathematics. [[Integer]]s and many other [[number system]]s form groups. Another common type is the [[symmetry group]] of a [[geometry|geometrical]] object, which consists of the set of transformations that leave the object unchanged; two transformations are combined by performing one after the other. Such symmetry groups, particularly the continuous [[Lie group]]s, play an important role in advanced theoretical [[physics]] and, to a lesser extent, in [[chemistry]]. [[history of group theory|Historically]], the concept of a group arose from the study of polynomial equations. Modern group theory studies parts and combinations of groups, such as [[subgroup]]s, [[quotient group]]s and [[simple group]]s. A particularly rich theory has been developed for [[finite group]]s, which culminated with the monumental [[classification of finite simple groups]]. {{TOClimit|limit=3}} == Definition and illustration == A group is a [[set (mathematics)|set]] equipped with an operation "•" which assigns to any two [[element (mathematics)|elements]] ''a'' and ''b'' another element denoted {{nowrap|''a'' • ''b''}}. To build a group, the operation is subject to certain requirements called ''group axioms''. Before giving the formal definition of group by specifying the four group axioms, the concept is illustrated with two key examples. ===First example: the integers=== The first example of a group is the set of [[integers]] '''Z''' containing the numbers :..., −4, −3, −2, −1, 0, 1, 2, 3, 4,&nbsp;... Together with the usual [[addition]] operation "+" they form what is probably the most familiar group. <ref>{{Harvard citations|last = Lang|year = 2005|loc = App. 2, p. 360|nb = yes}}</ref> The group axioms can be thought of as being modeled on the properties of the integers together with the addition operation. The following properties of numbers serve as a model for the abstract group axioms below. #For any two integers ''a'', ''b'', the [[sum]] ''a'' + ''b'' is also an integer. In other words, when adding two numbers, we do not leave the domain '''Z''' we are interested in, a property called ''closure''. #For all integers ''a'', ''b'' and ''c'', (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c''). This says that the two ways of calculating {{nowrap|''a'' + ''b'' + ''c''}} give the same result. This is referred to as the ''associativity'' of the addition. #If ''a'' is any integer, then 0 + ''a'' = ''a'' + 0 = ''a''. Therefore adding [[0 (number)|0]] amounts to no change, which is why 0 is called ''identity element''. #For each ''a'' in '''Z''', the element ''b'' = &minus;''a'' is an integer and satisfies ''a'' + ''b'' = ''b'' + ''a'' = 0. &minus;''a'' is called the (additive) ''inverse element'' of the integer ''a''. === Worked example: a symmetry group === {|align="right" style="margin: 1ex 1em 1ex 1em" | {|align="right" class="wikitable" style="text-align:center; margin: 0 auto 1ex auto" | [[Image:group D8 id.svg|140px]] <br /> id (keeping it as is) || [[Image:group D8 90.svg|140px]] <br /> r<sub>1</sub> (rotation by 90°) |- | [[Image:group D8 180.svg|140px]] <br /> r<sub>2</sub> (rotation by 180°) || [[Image:group D8 270.svg|140px]] <br /> r<sub>3</sub> (left rotation by 90°) |- | [[Image:group D8 fv.svg|140px]] <br /> f<sub>v</sub> (vertical flip) || [[Image:group D8 fh.svg|140px]] <br /> f<sub>h</sub> (horizontal flip) |- | [[Image:group D8 f13.svg|140px]] <br /> f<sub>d</sub> (diagonal flip) || [[Image:group D8 f24.svg|140px]] <br /> f<sub>c</sub> (counter-diagonal flip) |- | colspan=2 | Elements of the symmetry group.<br />The vertices are colored only to visualize the operations. |} |- | {| align="right" class="wikitable" style="text-align:center; margin: .5em auto 1em auto; width:40ex; height: 40ex" |+ [[Cayley table|group table]] |- !width="12%"| • !!width="11%"| id !!width="11%"| r<sub>1</sub> !!width="11%"| r<sub>2</sub> !!width="11%"| r<sub>3</sub> !!width="11%"| f<sub>v</sub> !!width="11%"| f<sub>h</sub> !!width="11%"| f<sub>d</sub> !!width="11%"| f<sub>c</sub> |- ! id |style="background:#FFDDDD; border-left: solid black 2px; border-top: solid black 2px"| id |style="background:#FFDDDD; border-top: solid black 2px"| r<sub>1</sub> |style="background:#FFDDDD; border-top: solid black 2px"| r<sub>2</sub> |style="background:#FFDDDD; border-right: solid black 2px; border-top: solid black 2px"| r<sub>3</sub> || f<sub>v</sub> || f<sub>h</sub> || f<sub>d</sub> |style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px; border-top: solid black 2px"| f<sub>c</sub> |- ! r<sub>1</sub> |style="background:#FFDDDD; border-left: solid black 2px;"| r<sub>1</sub> |style="background:#FFDDDD;"| r<sub>2</sub> |style="background:#FFDDDD;"| r<sub>3</sub> |style="background:#FFDDDD; border-right: solid black 2px;"| id || f<sub>c</sub> || f<sub>d</sub> || f<sub>v</sub> |style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;"| f<sub>h</sub> |- style="height:10%" ! r<sub>2</sub> |style="background:#FFDDDD; border-left: solid black 2px;"| r<sub>2</sub> |style="background:#FFDDDD;"| r<sub>3</sub> |style="background:#FFDDDD;"| id |style="background:#FFDDDD; border-right: solid black 2px;"| r<sub>1</sub> || f<sub>h</sub> || f<sub>v</sub> || f<sub>c</sub> |style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;"| f<sub>d</sub> |- style="height:10%" ! r<sub>3</sub> |style="background:#FFDDDD; border-bottom: solid black 2px;border-left: solid black 2px;"| r<sub>3</sub> |style="background:#FFDDDD; border-bottom: solid black 2px;"| id |style="background:#FFDDDD; border-bottom: solid black 2px;"| r<sub>1</sub> |style="background:#FFDDDD; border-right: solid black 2px; border-bottom: solid black 2px;"| r<sub>2</sub> || f<sub>d</sub> || f<sub>c</sub> || f<sub>h</sub> |style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px; border-bottom: solid black 2px"| f<sub>v</sub> |- style="height:10%" ! f<sub>v</sub> | f<sub>v</sub> || f<sub>d</sub> || f<sub>h</sub> || f<sub>c</sub>|| id || r<sub>2</sub> || r<sub>1</sub> || r<sub>3</sub> |- style="height:10%" ! f<sub>h</sub> | f<sub>h</sub> || f<sub>c</sub> || f<sub>v</sub> ||style="background:#DDDDFF;border:solid black 2px"| f<sub>d</sub> || r<sub>2</sub> || id || r<sub>3</sub> || r<sub>1</sub> |- style="height:10%" ! f<sub>d</sub> | f<sub>d</sub> || f<sub>h</sub> || f<sub>c</sub> || f<sub>v</sub> || r<sub>3</sub> || r<sub>1</sub> || id || r<sub>2</sub> |- style="height:10%" ! f<sub>c</sub> |style="background:#9DFF93; border-left: solid black 2px; border-bottom: solid black 2px; border-top: solid black 2px;" | f<sub>c</sub> |style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;" | f<sub>v</sub> |style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;" | f<sub>d</sub> |style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px; border-right: solid black 2px; " | f<sub>h</sub> || r<sub>1</sub> || r<sub>3</sub> || r<sub>2</sub> || id |- | colspan="9" style="text-align:left"| The elements id, r<sub>1</sub>, r<sub>2</sub>, and r<sub>3</sub> form a subgroup, highlighted in red. A left and right coset of this subgroup is highlighted in green and yellow, respectively. |} |} [[Group theory]] and the notion of a group concern much more general entities than numbers. [[Symmetry|Symmetries]], i.e. [[rotation]]s and [[reflection (mathematics)|reflections]] of a [[square (geometry)|square]] form a group called a [[dihedral group]], and denoted D<sub>4</sub>.<ref>{{Harvard citations|last = Herstein|year = 1975|loc = &#xa7;2.6, p. 54|nb = yes}}</ref> The following symmetries occur: :* rotations of the square by 90°, 180°, and 270°, denoted by r<sub>1</sub>, r<sub>2</sub> and r<sub>3</sub>, respectively; :* reflections about the vertical and horizontal middle line (f<sub>h</sub> and f<sub>v</sub>), or through the two [[diagonal]]s (f<sub>d</sub> and f<sub>c</sub>); :* the identity operation leaving everything unchanged, denoted id. Any two symmetries ''a'' and ''b'' can be [[Function composition|composed]], i.e. applied one after another. The result of performing first ''a'' and then ''b'' is written symbolically from right to left as :{{nowrap begin}}''b'' • ''a''{{nowrap end}} ("perform the symmetry ''b'' after performing the symmetry ''a''"). For example, rotating by 90° left (r<sub>3</sub>) and then flipping horizontally (f<sub>h</sub>) is the same as performing a reflection along the diagonal (f<sub>d</sub>). Using the above symbols, one has (highlighted in blue in the group table): :{{nowrap begin}}f<sub>h</sub> • r<sub>3</sub> = f<sub>d</sub>.{{nowrap end}}</li> Given this set of symmetries and the described operation, the group axioms can be understood as follows: <ol> <li> The ''closure'' axiom demands that the composition {{nowrap begin}}''b'' • ''a''{{nowrap end}} of any two symmetries ''a'' and ''b'' is still a symmetry. Another example is :{{nowrap begin}}r<sub>3</sub> • f<sub>h</sub> = f<sub>c</sub>,{{nowrap end}} i.e. rotating left by 270° after flipping horizontally equals flipping along the counter-diagonal. Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table. <li> The ''associativity'' constraint is the natural axiom to impose in order to make composing more than two symmetries well-behaved: given three elements ''a'', ''b'' and ''c'' of ''G'', there are two possible ways of computing "''a'' after ''b'' after ''c''". The requirement :{{nowrap begin}}(''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c''){{nowrap end}} means that composing ''a'' after ''b'', and calling this symmetry ''x'', then ''x'' after ''c'' is the same as ''a'' after ''y'', where ''y'' in turn is applying ''b'' after ''c''. For example, {{nowrap begin}} (f<sub>d</sub> • f<sub>v</sub>) • r<sub>2</sub> = f<sub>d</sub> • (f<sub>v</sub> • r<sub>2</sub>){{nowrap end}} can be checked using the [[Cayley table|group table]] at the right :{| |(f<sub>d</sub>||&nbsp;•&nbsp;||f<sub>v</sub>)||&nbsp;•&nbsp;||r<sub>2</sub>||&nbsp;=&nbsp;||r<sub>3</sub>||&nbsp;•&nbsp;||r<sub>2</sub>||&nbsp;=&nbsp;||r<sub>1</sub>, which equals |- |f<sub>d</sub>||&nbsp;•&nbsp;||(f<sub>v</sub>||&nbsp;•&nbsp;||r<sub>2</sub>)||&nbsp;=&nbsp;||f<sub>d</sub>||&nbsp;•&nbsp;||f<sub>h</sub>||&nbsp;=&nbsp;||r<sub>1</sub>. |} <li> The ''identity element'' is the symmetry id leaving everything unchanged: for any symmetry ''a'', performing id after ''a'' (or ''a'' after id) equals ''a'', in symbolic form, : {{nowrap begin}}id • ''a'' = ''a'',{{nowrap end}} : {{nowrap begin}}''a'' • id = ''a''.{{nowrap end}}</li> <li> An ''inverse element'' undoes the transformation of some other element. In the symmetry group example, every symmetry can be undone: each of the identity id, the flips f<sub>h</sub>, f<sub>v</sub>, f<sub>d</sub>, f<sub>c</sub> and the 180° rotation r<sub>2</sub> is its own inverse, because performing each one twice brings the square back to its original orientation. The rotations r<sub>3</sub> and r<sub>1</sub> are each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged. In symbols, :{{nowrap begin}}f<sub>h</sub> • f<sub>h</sub> = id,{{nowrap end}} :{{nowrap begin}}r<sub>3</sub> • r<sub>1</sub> = r<sub>1</sub> • r<sub>3</sub> = id.</li> </ol> ===Formal definition=== The formal group definition is an abstract formulation incorporating the essential features common to the integers and the above symmetry group: a group (''G'', •) is a [[set]] ''G'' with a [[binary operation]] • on ''G'' that satisfies the following four [[axiom]]s:<ref>{{Harvard citations|last = Herstein|year = 1975|loc = &#xa7;2.1, p. 27|nb = yes}}</ref> :{| | 1. ||width="150pt"| ''[[Closure (mathematics)|Closure]]''. || For all ''a'', ''b'' in ''G'', the result of ''a'' • ''b'' is also in ''G''. |- | 2. || ''[[Associativity]]''. || For all ''a'', ''b'' and ''c'' in ''G'', the equality (''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c'') holds. |- | 3. || ''[[Identity element]]''. || There exists a (unique) element ''e'' in ''G'' such that for all ''a'' in ''G'', one has ''e'' • ''a'' = ''a'' • ''e'' = ''a''. |- | 4. || ''[[Inverse element]]''. || For each ''a'' in ''G'', there exists an element ''b'' in ''G'' such that ''a'' • ''b'' = ''b'' • ''a'' = ''e'', where ''e'' is the identity element. |} The identity {{nowrap begin}}''a'' • ''b'' = ''b'' • ''a''{{nowrap end}} is not required to hold. The equality does hold in the first example of integers with addition, because {{nowrap begin}}''a'' + ''b'' = ''b'' + ''a''{{nowrap end}} for any two integers, but does not hold in the symmetry group example, because {{nowrap begin}}f<sub>h</sub> • r<sub>1</sub> = f<sub>c</sub>{{nowrap end}}, but {{nowrap begin}}r<sub>1</sub> • f<sub>h</sub> = f<sub>d</sub> and similar other examples.{{nowrap end}} == History == {{Main|History of group theory}} Historically, the group concept has evolved in several parallel threads.<ref>{{Harvard citations|nb = yes|last = Wussing|year = 2007}}</ref><ref>{{Harvard citations|last = Kleiner|year = 1986|nb = yes}}</ref><ref>{{Harvard citations|last = Smith|year = 1906|nb = yes}}</ref> One foundational root of group theory was the quest of solutions of [[polynomial equation]]s of degree higher than 4. [[Évariste Galois|Galois]], extending prior work of [[Paolo Ruffini|Ruffini]] and [[Joseph-Louis Lagrange|Lagrange]], introduced groups expressing symmetry features of the solutions of such equations, thus giving a criterion for the [[solution (disambiguation)|solvability]] of the equation in question. These groups are nowadays called [[Galois group]]s.<ref>{{Harvard citations|last = Galois|year = 1908|nb = yes}}</ref><ref>{{Harvard citations|last = Kleiner|year = 1986|loc = p. 202|nb = yes}}</ref> These groups can be viewed as consisting of certain [[permutation]]s of these solutions. Similar groups are generally called [[permutation group]]s, a concept investigated in particular by [[Augustin Louis Cauchy|Cauchy]]. [[Arthur Cayley|Cayley's]] ''On the theory of groups, as depending on the symbolic equation θ<sup>n</sup> = 1'' (1854) gives the first abstract definition of [[finite group]]s. Secondly, the systematic use of groups in geometry, mainly in the guise of [[symmetry group]]s, was initiated by [[Felix Klein|Klein's]] 1872 [[Erlangen program]].<ref>{{Harvard citations|last = Wussing|year = 2007|loc = &#xa7;III.2|nb = yes}}</ref> The study of what are now called [[Lie group]]s started systematically in 1884 with [[Sophus Lie]]. The third root of group theory was [[number theory]]. Certain [[abelian group]] structures had been implicitly used in [[number theory|number-theoretical]] work by [[Carl Friedrich Gauss|Gauss]], and more explicitly by [[Kronecker]].<ref>{{Harvard citations|last = Kleiner|year = 1986|loc = p. 204|nb = yes}}</ref> Early attempts to prove [[Fermat's last theorem]] were led to a climax by [[Ernst Kummer|Kummer]] by introducing [[class group|groups describing factorization]] into [[prime number]]s.<ref>{{Harvard citations|last = Wussing|year = 2007|loc = &#xa7;I.3.4|nb = yes}}</ref> The convergence of these various sources into a uniform theory of groups started with [[Camille Jordan|Jordan's]] ''Traité des substitutions et des équations algébriques'' ([[1870]]) and [[Walther von Dyck|von Dyck]] ([[1882]]) who first defined a group in the full abstract sense of this article. The early 20th century's group theory encompassed roughly the content of the basic concepts (see [[#Basic concepts in group theory|below]]). Group theory subsequently grew both in depth and in breadth, branching out into areas such as [[algebraic group]]s, [[group extension]]s, and [[representation theory]].<ref>{{Harvard citations|last = Curtis |year = 2003|nb = yes}}</ref> Starting in the 1950s, in a huge collaborative effort, group theorists succeeded to [[classification of finite simple groups|classify]] all finite [[simple group]]s in 1982. Completing and simplifying the proof of the classification are areas of active research.<ref>{{Harvard citations|last = Aschbacher|year = 2004|nb = yes}}</ref> ==First consequences of the group axioms == [[Elementary group theory]] is concerned with basic facts about general groups, as opposed for example to the more involved study of groups via their [[group representation|representations]].<ref name=FultonHarris>{{Harvard citations|last1 = Fulton|last2 = Harris|year = 1991|nb = yes}}</ref> These facts are usually direct consequences of the group definition — obtained by invoking the axioms a few times.<ref>{{Harvard citations|last = Ledermann|year = 1953|loc = &#xa7;1.2, pp. 4&ndash;5|nb = yes}}</ref> For example, [[Mathematical induction|repeated]] applications of the associativity axiom show that the unambiguity of :''a'' • ''b'' • ''c'' = (''a'' • ''b'') • ''c'' = ''a'' • (''b'' • ''c'') generalizes to more than three factors. Therefore parentheses are usually omitted in such expressions.<ref>{{Harvard citations|nb = yes|last = Ledermann|year = 1973|loc = &#xa7;I.1, p. 3}}</ref> ===Uniqueness of identity element and inverses=== Though the uniqueness of the identity and inverse elements is not required by the group axioms, it is a consequence of them. Therefore it is customary to speak of ''the'' identity, and ''the'' inverse of ''a''.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.1, p. 17}}</ref> The following proof of the latter fact shows the flavor of elementary group theory: suppose given two inverses ''l'' and ''r'' of a fixed element ''a''. Then :''l''&nbsp;=&nbsp;''l'' • ''e'' = ''l'' • (''a'' • ''r'') = (''l'' • ''a'') • ''r'' = ''e'' • ''r'' = ''r''. Moreover, in a group, knowing only that ''n''&nbsp;•&nbsp;''m''&nbsp;=&nbsp;''e'' (or {{nowrap begin}}''m'' • ''n'' = ''e''){{nowrap end}} suffices to conclude that ''n'' is the inverse element of ''m''.{{cref|a}} The inverse of a product is the product of the inverses in the opposite order: (''a''&nbsp;•&nbsp;''b'')<sup>−1</sup> = ''b''<sup>−1</sup> • ''a''<sup>−1</sup>. To prove this it is enough (by the previous remark, applied to ''m''=''a''&nbsp;•&nbsp;''b'' and ''n''=''b''<sup>&minus;1</sup>&nbsp;•&nbsp;''a''<sup>&minus;1</sup>) to prove the identity (''a''&nbsp;•&nbsp;''b'')&nbsp;•&nbsp;(''b''<sup>&minus;1</sup>&nbsp;•&nbsp;''a''<sup>&minus;1</sup>)&nbsp;=&nbsp;''e'': ::{| |(''a''&nbsp;•&nbsp;''b'')&nbsp;•&nbsp;(''b''<sup>&minus;1</sup>&nbsp;•&nbsp;''a''<sup>&minus;1</sup>)|| = || ((''a''&nbsp;•&nbsp;''b'')&nbsp;•&nbsp;''b''<sup>&minus;1</sup>&nbsp;)&nbsp;•&nbsp;''a''<sup>&minus;1</sup>|| (associativity) |- | || = || (''a''&nbsp;•&nbsp;(''b''&nbsp;•&nbsp;''b''<sup>&minus;1</sup>))&nbsp;•&nbsp;''a''<sup>&minus;1</sup> || (associativity) |- | || = || (''a''&nbsp;•&nbsp;''e'')&nbsp;•&nbsp;''a''<sup>&minus;1</sup> || (definition of inverse) |- | || = || ''a''&nbsp;•&nbsp;''a''<sup>&minus;1</sup> || (definition of identity element) |- | || = || ''e'' || (definition of inverse) |} Because inverse elements are unique, the two inverses (''a''&nbsp;•&nbsp;''b'')<sup>−1</sup> and ''b''<sup>&minus;1</sup>&nbsp;•&nbsp;''a''<sup>&minus;1</sup> must be equal. ===Division=== In groups, it is possible to perform [[division (mathematics)|division]]: given elements ''a'' and ''b'' of the group ''G'', there is exactly one solution ''x'' in ''G'' to the [[equation]] {{nowrap begin}}''x'' • ''a'' = ''b''{{nowrap end}}.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.1, p. 17}}</ref> In fact, right multiplication of the equation by ''a''<sup>&minus;1</sup> gives the solution {{nowrap begin}}''x'' = ''x'' • ''a'' • ''a''<sup>&minus;1</sup> = ''b'' • ''a''<sup>&minus;1</sup>{{nowrap end}}. Similarly there is exactly one solution ''y'' in ''G'' to the equation {{nowrap begin}}''a'' • ''y'' = ''b''{{nowrap end}}, namely {{nowrap begin}}''y'' = ''a''<sup>&minus;1</sup> • ''b''{{nowrap end}}. In general, ''x'' and ''y'' need not agree. ===Variants of the definition=== {{Group-like structures}} Some definitions of a group use seemingly weaker conditions for identity and inverse elements. For instance, the axioms may be weakened to assert only the existence of a [[left identity]] and a [[left inverse]] for every element. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one above.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = &#xa7;I.2, p. 7}}</ref> Strictly speaking the closure axiom is already implied by the condition that • be a binary operation on ''G''. Many authors (such as {{Harvard citations|txt = yes|last = Lang|year = 2002}}) therefore omit this axiom. In [[abstract algebra]], more general structures arise by relaxing some of the axioms defining a group, listed in the table.<ref name="MacLane">{{Harvard citations|nb = yes|last = Mac Lane|year = 1998}}</ref><ref>{{Harvard citations|nb = yes|last1 = Denecke|last2 = Wismath|year = 2002}}</ref><ref>{{Harvard citations|nb = yes|last1 = Romanowska|last2 = Smith|year=2002}}</ref> For example, eliminating the requirement that every element have an inverse, then the resulting algebraic structure is called a [[monoid]]. The integers under multiplication ('''Z''', •) are an example (see below). [[Groupoid]]s are similar to groups except that the composition ''a'' • ''b'' need not be defined for all ''a'' and ''b''. They arise in the study of more involved kinds of symmetries, often in [[topology|topological]] and [[analysis|analytical]] structures, e.g. the [[fundamental groupoid]]. === Notations === {| align="left" class="infobox" style="text-align:center" |- align="center" | style="border-bottom: 2px solid #303060" colspan=5| Customary notations for group operations |- | || operation || identity || inverse of ''a'' |- | [[additive group|additive notation]] | + || 0 || &minus;''a'' |- | [[multiplicative group|multiplicative notation]] | *, •, &times; || 1 || ''a''<sup>&minus;1</sup> |- | [[automorphism group|notation related to functions]] | ∘ || id, 1 || ''a''<sup>–1</sup> |} The notation for groups often depends on the context and the nature of the group operation. There is a tendency to denote [[abelian group]]s additively, whereas non-abelian groups are often written multiplicatively.<ref>{{Harvard citations|nb = yes|last = Artin|year = 1991|loc = &#xa7;2.1}}</ref> In many situations, there is only one possible (or reasonable) group operation on a given set, therefore it is very common to drop the operation symbol and leave it to the reader to know the context and the group operation. For example the groups ([[cyclic group|'''Z'''<sub>''n''</sub>]], +) and ('''F'''<sub>''q''</sub><sup>&times;</sup>, •), the multiplicative group of nonzero elements in the [[finite field]] '''F'''<sub>''q''</sub> are commonly denoted '''Z'''<sub>''n''</sup> and '''F'''<sub>''q''</sub><sup>&times;</sup>, since only one of the two [[ring (mathematics)|ring]] operations makes these sets into a group.{{cref|b}} == Basic concepts == <div class=dablink>The following sections use [[table of mathematical symbols|mathematical symbols]] such as X = ''{''x'', ''y'', ''z''}'' to denote a [[set (mathematics)|set]] X containing [[element (mathematics)|elements]] x, y, and z, or alternatively x ''∈'' X to restate that x is an element of X. The notation {{nowrap|f : X ''&rarr;'' Y}} means a [[function (mathematics)|function]] assigning to every element of X an element of Y.</div> {{Basic notions in group theory}} The [[glossary of group theory|arsenal of basic group theory]] comprises various methods to handle groups. The structure of groups can be understood by breaking them into pieces called subgroups and quotient groups. Combining them into larger groups yields direct and semidirect products. An equally important technique, fundamental to group theory, is comparing groups using homomorphisms. A particularly well-understood class of groups are the abelian groups. These basic concepts form the standard introduction to groups (see, for example, the books of {{Harvard citations|last = Lang|year = 2002|year2 = 2005|txt = yes}} and {{Harvard citations|last = Herstein|year = 1996|year2 = 1975|txt = yes}}). ===Subgroups=== {{Main|Subgroup}} Informally, a ''subgroup'' is a group contained in a bigger one. More precisely, a [[subset]] ''H'' of ''G'' is called a subgroup if the restriction of • to ''H'' is a group operation on ''H''.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.1, p. 19}}</ref> In other words, the identity element of ''G'' is contained in ''H'', and whenever ''g'' and ''h'' are in ''H'', then so is {{nowrap|''g'' • ''h''}} and ''g''<sup>&minus;1</sup>. In the example above, the rotations {{nowrap begin}}''R'' = {id, r<sub>1</sub>, r<sub>2</sub>, and r<sub>3</sub>}{{nowrap end}} constitute a subgroup, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the rotation in the opposite direction. The [[subgroup test]] is a necessary and sufficient condition for a subset ''H'' of a group ''G'' to be a subgroup: it is sufficient to check that {{nowrap|''g''<sup>−1</sup>''h'' ∈ ''H''}} for all elements {{nowrap begin}}''g'', ''h'' ∈ ''H''{{nowrap end}}. Knowing [[lattice of subgroups|the subgroups]] is important to understand the group as a whole.{{cref|c}} Given any subset ''S'' of a group ''G'', the subgroup ''generated by S'' consists of products of elements of ''S'' and their inverses. It is also the smallest subgroup of ''G'' containing ''S''.<ref>{{Harvard citations|nb = yes|last = Ledermann|year = 1973|loc = &#xa7;II.12, p. 39}}</ref> In the introductory example above, the subgroup generated by r<sub>2</sub> and f<sub>v</sub> consists of these two elements, the identity element id and {{nowrap begin}}f<sub>h</sub> = f<sub>v</sub> • r<sub>2</sub>{{nowrap end}}. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup. ===Cosets=== {{Main|Coset}} A subgroup ''H'' defines left and right ''cosets'', which can be thought of as [[translation (geometry)|translations]] of ''H'' by arbitrary group elements ''g''. In symbolic terms, the ''left'' and ''right coset'' of ''H'' containing ''g'' are :{{nowrap begin}}''gH'' = {''gh'', ''h'' &isin; ''H''}{{nowrap end}} and {{nowrap begin}}''Hg'' = {''hg'', ''h'' &isin; ''H''}{{nowrap end}}, respectively.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.4, p. 41}}</ref> The cosets of any subgroup ''H'' form a [[Partition of a set|partition]] of the elements of ''G''; that is, two left cosets are either equal or have an [[empty set|empty]] [[Intersection (set theory)|intersection]].<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = &#xa7;I.2, p. 12}}</ref> The same holds true of the right cosets of ''H''. Left and right cosets of ''H'' may or may not be equal. If they are, i.e. for all ''g'' in ''G'', ''gH''&nbsp;=&nbsp;''Hg'', then ''H'' is said to be a ''[[normal subgroup]]''. One may then speak simply of the set of cosets of ''N''. In D<sub>4</sub>, the introductory symmetry group, the (left) cosets of the subgroup ''R'' consisting of the rotations are either equal to ''R'', if ''g'' is an element of ''R'' itself, or otherwise equal to {{nowrap begin}}''U'' = f<sub>v</sub>''R'' = {f<sub>v</sub>, f<sub>d</sub>, f<sub>h</sub>, f<sub>c</sub>}{{nowrap end}} (highlighted in green). This subgroup is also normal, because {{nowrap begin}}f<sub>v</sub>''R'' = ''U'' = ''R''f<sub>v</sub>{{nowrap end}} and similarly for any other group element instead of f<sub>v</sub>. ===Quotient groups=== {| align="right" class="wikitable" style="text-align:center;" |- !width="50px"| • !width="33%"| R !width="33%"| U |- ! ''R'' | ''R'' || ''U'' |- ! ''U'' | ''U'' || ''R'' |- |} {{Main|Quotient group}} The ''quotient group'' or ''factor group'' :''G'' / ''N'' = {''gN'', ''g'' &isin; ''G''}, "''G'' modulo ''N''" treats the cosets of a normal subgroup ''N'' as a group.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.4, p. 45}}</ref> The group operation on this set (sometimes called coset multiplication, or coset addition) behaves in the nicest way possible: {{nowrap begin}}(''gN'') • (''hN'') = (''gh'')''N''{{nowrap end}} for all ''g'' and ''h'' in ''G''. The coset {{nowrap begin}}''eN '' = ''N''{{nowrap end}} itself serves as the identity in this group, and the inverse of ''Ng'' in the quotient group is {{nowrap begin}}(''gN'')<sup>&minus;1</sup> = (''g''<sup>&minus;1</sup>)''N''.{{nowrap end}}{{cref|d}} The elements of the corresponding quotient group {{nowrap|D<sub>4</sub> / ''R''}} are ''R'' itself, which represents the identity, and {{nowrap begin}}''U'' = f<sub>v</sub>''R''{{nowrap end}}. The group operation on the quotient is shown at the right. For example, {{nowrap begin}}''U'' • ''U'' = f<sub>v</sub>''R'' • f<sub>v</sub>''R'' = (f<sub>v</sub> • f<sub>v</sub>)''R'' = ''R''{{nowrap end}}. Quotient and subgroups together form a way of describing every group by its ''[[presentation of a group|presentation]]'': any group is the quotient of the [[free group]] over the ''[[Generating set of a group|generators]]'' of the group, quotiented by the subgroup of ''relations''. The dihedral group D<sub>4</sub>, for example, can be generated by two elements ''r'' and ''f'' (for example, ''r''&nbsp;=&nbsp;r<sub>1</sub>, the right rotation and ''f''&nbsp;=&nbsp;f<sub>v</sub> the vertical (or any other) flip), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations :''r ''<sup>4</sup>&nbsp;=&nbsp;''f ''<sup>2</sup>&nbsp;=&nbsp;(''rf'')<sup>2</sup>&nbsp;=&nbsp;1,<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = &#xa7;I.2, p. 9}}</ref> the group is completely described. A presentation of a group can also be used to construct the [[Cayley graph]], a graphical device showing certain features of [[discrete group]]s. ===Products=== {{Main|Direct product|Semidirect product}} Taking subgroups and quotients of a given group ''G'' tends to reduce the size of ''G''.{{cref|e}} Several group constructions reverse this direction, i.e. given two groups, one constructs bigger groups, such as the ''direct product'' ''G''×''H'' of the two. (Here, "product" has a slightly different meaning than the product of elements in a group.) It consists of [[ordered pair|pairs]] (''g'', ''h''), ''g'' in ''G'' and ''h'' in ''H'', with the group operation :(''g''<sub>1</sub>, ''h''<sub>1</sub>) • (''g''<sub>2</sub>, ''h''<sub>2</sub>) = (''g''<sub>1</sub> • ''g''<sub>2</sub>, ''h''<sub>1</sub> • ''h''<sub>2</sub>).<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = &#xa7;I.2}}</ref> A further generalization of the direct product of two groups is the ''semidirect product''; it allows for the twisting of the group operation on one factor. The group of symmetries of the square (described above) is a semidirect product of the subgroup ''R'' consisting of rotations with the corresponding quotient (generated by a reflection). ===Group homomorphisms=== {{Main|Group homomorphism}} ''Group homomorphisms''{{cref|f}} are [[function (mathematics)|functions]] that preserve the structure of the groups in question. The structure being determined by the group operation, this is made formal by requiring :{{nowrap begin}}''a''(''g'' • ''k'') = ''a''(''g'') • ''a''(''k'').{{nowrap end}} for a function {{nowrap begin}}''a'': ''G'' → ''H''{{nowrap end}} and any two elements ''g'', ''k'' in ''G''. This requirement ensures that {{nowrap begin}}''a''(1<sub>''G''</sub>) = 1<sub>''H''</sub>{{nowrap end}}, and also {{nowrap begin}}''a''(''g'')<sup>&minus;1</sup> = ''a''(''g''<sup>&minus;1</sup>){{nowrap end}} for all ''g'' in ''G'', so the additional data from the group axioms are respected, as well.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.3, p. 34}}</ref> Two groups ''G'' and ''H'' are called [[isomorphism|isomorphic]] if there exist group homomorphisms {{nowrap begin}}''a'': ''G'' &rarr; ''H''{{nowrap end}} and {{nowrap begin}}''b'': ''H'' &rarr; ''G''{{nowrap end}}, such that applying the two functions [[function composition|one after another]] (in the two possible ways) equal the [[identity function]] of ''G'' and ''H'', respectively, i.e. {{nowrap begin}}''a''(''b''(''h'')) = ''h''{{nowrap end}}, and {{nowrap begin}}''b''(''a''(''g'')) = ''g''{{nowrap end}} for any ''g'' in ''G'' and ''h'' in ''H''. From an abstract point of view, isomorphic groups carry practically the same information. For example, proving that {{nowrap begin}}''g'' • ''g'' = 1{{nowrap end}} for some element ''g'' of ''G'' is [[Logical equivalence|equivalent]] to proving that {{nowrap begin}}''a''(''g'') • ''a''(''g'') = 1,{{nowrap end}} because applying ''a'' to the first equality yields the second, and applying ''b'' to the second gives back the first. The [[category of groups]] is an abstract framework containing groups and group homomorphisms.<ref name="MacLane"/> For any group homomorphism ''a'': ''G'' → ''H'', the [[kernel (algebra)|kernel]] {{nowrap begin}}ker ''a'' = {''g'' in ''G'' : ''a''(''g'') = 1<sub>''H''</sub>}{{nowrap end}} is the set of elements in ''G'' which are mapped to the identity in ''H''. The kernel and [[Image (mathematics)|image]] {{nowrap begin}}''a''(''G'') = {''a''(''g''), ''g'' ∈ ''G''}{{nowrap end}} of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The [[isomorphism theorem|First Isomorphism Theorem]] states that the [[Image (mathematics)|image]] of a group homomorphism, ''a''(''G'') is isomorphic to the quotient group ''G''/ker ''a''.{{cref|g}} ===Abelian groups=== {{Main|Abelian group}} A group ''G'' is said to be [[abelian group|''abelian'']] (in honor of [[Niels Henrik Abel]]), or ''commutative'', if the operation satisfies the [[commutative law]] :{{nowrap begin}}''a'' • ''b'' = ''b'' • ''a''{{nowrap end}} for all group elements ''a'' and ''b''.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = &#xa7;I.1, p. 4}}</ref> If not, the group is called ''non-abelian'' or ''non-commutative''. The study of 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.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = &#xa7;I.5, p. 26, 29}}</ref> As noted in the first section, the group of symmetries of the square (discussed above) is non-abelian. However, the subgroup {{nowrap begin}}''R'' = {id, r<sub>1</sub>, r<sub>2</sub>, r<sub>3</sub>}{{nowrap end}} consisting of the rotations, as well as the quotient with respect to this subgroup, are abelian. This fact is reflected in the semi-direct product structure of this group (see above). ====Cyclic groups==== [[Image:Cyclic group.svg|right|thumb|150px|The 6<sup>th</sup> complex roots of unity form a cyclic group. ''&zeta;'' is a primitive element, but ''&zeta;''<sup>2</sup> is not, because the odd powers of ''&zeta;'' are not a power of ''&zeta;''<sup>2</sup>.]] {{Main|Cyclic group}} A ''cyclic group'' is a group all of whose elements are [[power (mathematics)|powers]] (when the group operation is written multiplicatively) or multiples (written additively) of a particular element ''a''.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.1, p. 22}}</ref> In multiplicative notation, the elements of the group are: :..., ''a''<sup>−3</sup>, ''a''<sup>−2</sup>, ''a''<sup>−1</sup>, ''a''<sup>0</sup> = ''e'', ''a'', ''a''<sup>2</sup>, ''a''<sup>3</sup>, ..., where ''a''<sup>2</sup> means ''a''•''a'', and ''a<sup>&minus;3</sup>'' stands for ''a''<sup>&minus;1</sup>•''a''<sup>&minus;1</sup>•''a''<sup>&minus;1</sup>=(''a''•''a''•''a'')<sup>&minus;1</sup> etc.{{cref|h}} Such an element ''a'' is called a generator or a [[Primitive root modulo n|primitive element]] of the group. The eponym for this class of groups is the group of [[root of unity|''n''-th complex roots of unity]], given by [[complex number]]s ''ω'' satisfying {{nowrap begin}}''ω''<sup>''n''</sup> = 1{{nowrap end}} (and whose operation is multiplication).<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.2, p. 26}}</ref> Any cyclic group with ''n'' elements is isomorphic to this group. An infinite cyclic group is isomorphic to ('''Z''', +), the group of integers under addition introduced above.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;II.1, p. 22 (example 11)}}</ref> As these two prototypes are both abelian, so is any cyclic group. ==Examples and applications== [[Image:Wallpaper group-cm-6.jpg|thumb|right|150px|A periodic wallpaper gives rise to a [[wallpaper group]].]] [[Image:Fundamental group.svg|left|200px|thumb|The fundamental group of a plane minus a point (bold) consists of loops in this area. The blue loop is considered [[null-homotopic|irrelevant]], because it can be [[homotopy|shrunk]] to a point. The presence of the hole prevents the orange loop from being shrunk. This way, the fundamental group detects the hole.]] {{Main|Examples of groups|Applications of group theory}} ''Examples'' and ''applications of groups'' abound. A starting point is the group '''Z''' 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. A major theme in contemporary mathematics is to study given objects by [[functor|associating]] groups to them. For example, [[Henri Poincaré]] founded what is now called [[algebraic topology]] by introducing the [[fundamental group]].<ref>{{Harvard citations|nb = yes|last = Hatcher|year = 2002|loc = Chapter I, p. 30}}</ref> By means of this connection, [[Glossary of topology|topological properties]] such as [[Neighbourhood (mathematics)|proximity]] and [[continuous function|continuity]] translate into properties of groups.{{cref|i}} In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.{{cref|j}} In a similar vein, [[geometric group theory]] employs geometric concepts, for example in the study of [[hyperbolic group]]s.<ref>{{Harvard citations|nb = yes|last1 = Coornaert|last2 = Delzant|year = 1990}}</ref> Further branches crucially applying groups include [[algebraic geometry]] and [[number theory]].<ref>For example, [[class group]]s and [[Picard group]]s; see {{Harvard citations|txt = yes|last = Neukirch|year = 1999}}, in particular &#xa7;&#xa7;I.12 and I.13.</ref> In addition to the above theoretical applications, many practical applications of groups exist. [[Cryptography]] stakes on the combination of the abstract group theory approach together with [[algorithm|algorithmical]] knowledge obtained in [[computational group theory]], in particular when implemented for finite groups.<ref>{{Harvard citations|nb = yes|last = Seress|year = 1997}}</ref> Applications of group theory are by no means restricted to mathematics. Other sciences such as [[physics]], [[chemistry]] or [[computer science]] benefit from the abstract group concept, as well. === Numbers === ==== Integers ==== The integers '''Z''' under addition form a group ('''Z''', +), described above. In addition to merely being a group, this group is also abelian because :''a'' + ''b'' = ''b'' + ''a'' ([[commutativity]] of addition). The integers, with the operation of [[multiplication]] instead of addition, denoted ('''Z''',&nbsp;&middot;) do ''not'' form a group. The closure, associativity and identity axioms are satisfied, but inverses do not exist: for example, {{nowrap begin}}''a'' = 2{{nowrap end}} is an integer, but the only solution to the equation ''a&middot;b'' = 1 in this case is ''b'' = 1/2, which is a rational number, but not an integer. Hence not every element of '''Z''' has a (multiplicative) inverse.{{cref|k}} ==== Rationals ==== The wished-for existence of a multiplicative inverses suggests considering [[fraction (mathematics)|fractions]] :<math>\frac{a}{b}</math>. Fractions of integers (with ''b'' nonzero) are known as [[rational number]]s.{{cref|l}} The set of all such fractions is commonly denoted '''Q'''. There is still a minor obstacle for {{nowrap|('''Q''', &middot;),}} the rationals with multiplication, being a group: since the rational number [[0 (number)|0]] does not have a multiplicative inverse, ('''Q''',&nbsp;&middot;) is still not a group. However, the set of all ''nonzero'' rational numbers {{nowrap begin}}'''Q''' \ {0} = {''q'' ∈ '''Q''', ''q'' ≠ 0}{{nowrap end}} ''does'' form an abelian group under multiplication, denoted {{nowrap|('''Q''' \ {0}, &middot;)}}.{{cref|m}} 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 ''a''/''b'' is ''b''/''a'', therefore the axiom of the inverse element is satisfied. The rational numbers (including 0) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called [[ring (mathematics)|rings]] and – if division is possible, such as in '''Q''' – [[field (mathematics)|fields]], which occupy a central position in [[abstract algebra]]. Group theoretic arguments therefore underlie parts of the theory of those entities.{{cref|n}} ===== Cyclic multiplicative groups ===== In ('''Q'''&nbsp;\&nbsp;{0},&nbsp;&middot;), there are cyclic subgroups :''G'' = {''a''<sup>''n''</sup>, ''n'' &isin; '''Z'''} where ''a<sup>n</sup>'' is the ''n''-th [[exponentiation]]s of the primitive element ''a'' of that group.<ref>This example is from {{Harvard citations|txt = yes|last = Lang|year = 2005|loc = &#xa7;II.1, p. 22}}</ref> For example, if ''a'' is 2 then :G&nbsp;=&nbsp;{..., 2<sup>&minus;2</sup>, 2<sup>&minus;1</sup>, 2<sup>0</sup>, 2<sup>1</sup>, 2<sup>2</sup>, ...} = {..., 0.25, 0.5, 1, 2, 4, ...}. This group is an example of a [[free abelian group]] of [[rank of an abelian group|rank]] one: the rank is one, because ''G'' is generated by one element (''a'' or equivalently ''a''<sup>&minus;1</sup>) and the freeness refers to the fact that no [[presentation of a group|relations]] between the powers of this generator occur. Therefore, ''G'', is isomorphic to the (additive) group of integers ('''Z''',&nbsp;+) above. This example shows that distinguishing between additive and multiplicative groups is merely a matter of notation. ==== Nonzero integers modulo a prime ==== For any [[prime number]] ''p'', [[modular arithmetic]] furnishes the [[multiplicative group of integers modulo n|multiplicative group of integers modulo ''p'']].<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = Chapter VII}}</ref> Its elements are integers not divisible by ''p'', considered [[Modular arithmetic|modulo]] ''p''. The latter means that two numbers are considered equivalent if their [[difference (mathematics)|difference]] is divisible by ''p''. For example, if ''p''=5, then 4·3=2 in this group, because the usual product 12 is equivalent to 2, for 5 divides {{nowrap begin}}12 &minus; 2 = 10{{nowrap end}}, denoted :12 ≡ 2 ([[modulo|mod]] 5). The primality of ''p'' ensures that the product of two integers neither of which is divisible by ''p'' is not divisible by ''p'' either, hence the indicated set of classes is closed under multiplication.{{cref|o}} The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Finally, the inverse element axiom requires that given an integer ''a'' not divisible by ''p'', there exists an integer ''b'' such that :''a''&nbsp;&middot;&nbsp;''b''&nbsp;≡&nbsp;1 (mod ''p''), i.e. ''p'' divides the difference {{nowrap|''a'' &middot; ''b'' &minus; 1}}. The inverse ''b'' can be found by using that the [[greatest common divisor]] {{nowrap|gcd(''a'', ''p'')}} equals 1.<ref>{{Harvard citations|nb = yes|last = Rosen|year = 1984|loc = p. 54 (Theorem 2.1)}}</ref> Hence all group axioms are fulfilled. Actually, this example is similar to ('''Q'''\{0}, &middot;) above, because it turns out to be the multiplicative group of nonzero elements in '''F'''<sub>''p''</sub>, denoted '''F'''<sub>''p''</sub><sup>&times;</sup>.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2005|loc = &#xa7;VIII.1, p. 292}}</ref> === Finite groups === {{Main|Finite group}} A group is called ''finite'' if it has [[finite set|finitely many elements]]. The number of elements is called ''[[order (group theory)|order]]'' of the group ''G''.<ref>{{Harvard citations|nb = yes|last1 = Kurzweil|last 2= Stellmacher|year = 2004}}</ref> An important class are the ''[[symmetric group]]s'' ''S''<sub>''N''</sub>, the groups of [[permutation]]s of ''N'' letters. For example, the symmetric group on 3 letters [[dihedral group of order 6|''S''<sub>3</sub>]] is the group consisting of all possible swaps of the three letters ''ABC'', i.e. contains the elements ''ABC'', ''ACB'', ..., up to ''CBA'', in total 6 (or 3 [[factorial]]) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group ''S''<sub>''N''</sub> for a suitable ''N'' ([[Cayley's theorem]]). Parallel to the group of symmetries of the square above, ''S''<sub>3</sub> can also be interpreted as the group of symmetries of an [[equilateral triangle]]. The ''order of an element a'' in a group ''G'' is the least positive integer ''n'' such that ''a<sup>n</sup>&nbsp;=&nbsp;e'', where ''a<sup>n</sup>'' represents <math>\underbrace{a \cdot \ldots \cdot a}_n</math>, i.e. application of the operation • to ''n'' copies of the value ''a''. (If • represents multiplication, then ''a''<sup>''n''</sup> corresponds to the ''n''<sup>th</sup> power of ''a''.) If no such ''n'' exists, then the order of ''a'' is said to be infinity. The order of an element is the same as the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: [[Lagrange's theorem (group theory)|Lagrange's Theorem]] states that for a finite group ''G'' the order of any (necessarily finite) subgroup ''H'' [[divisor|divides]] the order of ''G''. The [[Sylow theorems]] give a partial converse, by asserting the existence of [[p-group|subgroups whose order is any prime power]] dividing the order of the group. The [[dihedral group]] (discussed above) is a finite group of order 8. The order of r<sub>1</sub> is 4, as is the order of the subgroup ''R'' it generates (see above). The order of the reflection elements f<sub>v</sub> etc. is 2. Both orders divide 8, as predicted by Lagrange's Theorem. The groups '''F'''<sub>''p''</sub><sup>&times;</sup> above have order ''p''&minus;1. The latter groups are crucial to [[public-key cryptography]].{{cref|p}} ==== Classification of finite simple groups ==== Given any mathematical notion, mathematicians often aim for a complete [[classification theorems|classification]] (or list) of them. In the context of finite groups, this aim quickly leads to very deep mathematics. According to Lagrange's theorem, finite groups of order ''p'', a prime number, are necessarily cyclic (abelian) groups '''Z'''<sub>''p''</sub>. Groups of order ''p''<sup>2</sup> can also be shown to be abelian, a statement which does not generalize to order ''p''<sup>3</sup>, as the non-abelian group D<sub>4</sub> of order 8=2<sup>3</sup> above shows.<ref>{{Harvard citations|nb = yes|last = Artin|year = 1991|loc = Theorem 6.1.14}}. See also {{Harvard citations|txt = yes|last = Lang|year = 2002|loc = p. 77}} for similar results.</ref> [[Computer algebra system]]s can be used to [[List of small groups|list ''small'' groups]], but there is no classification of ''all'' finite groups.{{cref|q}} An intermediate step is the [[classification of finite simple groups]].{{cref|r}} A nontrivial group is called ''[[simple group|simple]]'' if its only normal subgroups are the [[trivial group]] and the group itself.{{cref|s}} The [[Jordan-Hölder theorem]] exhibits simple groups as the building blocks for all finite groups.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = &#x7a;I.3, p. 22}}</ref> [[List of finite simple groups|Listing]] all finite simple groups was a major achievement in contemporary group theory. The [[monstrous moonshine]] conjectures, proven by 1998 [[Fields Medal]] winner [[Richard Borcherds]], provide a surprising and deep connection between the largest finite simple [[sporadic group]], called the [[monster group]], and [[modular function]]s and [[string theory]].<ref>{{Harvard citations|nb = yes|last = Ronan|year = 2007}}</ref> ===Symmetry groups=== [[Image:Uniform tiling 73-t2 colored.png|right|150px|thumb|The [[(2,3,7) triangle group]], a hyperbolic group, acts on this [[Tessellation|tiling]] of the [[hyperbolic plane]].]] {{Main|Symmetry group}} ''Symmetry groups'' are groups consisting of [[symmetry|symmetries]] of given mathematical objects – be they of geometric nature, such as introductory symmetry group of the square, or of algebraic nature, such as [[polynomial equation]] and their solutions.<ref>{{Harvard citations|nb = yes|last = Weyl|year = 1952}}</ref> Conceptually, group theory can be thought of as the study of symmetry.{{cref|t}} Symmetries greatly [[symmetry in mathematics|simplify the study]] of [[geometry|geometrical]] or [[analysis|analytical objects]]. This remark is formalized and exploited using the notion of [[group action]]s, which means that every group element performs some operation on another mathematical object, in a way compatible to the group structure. This way, the group leaves its footprints on the mathematical object. In the example at the right, a group element of order 7 acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). 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, or in [[CD player]]s.<ref>{{Harvard citations|nb = yes|last = Welsh|year = 1989}}</ref> Further applications include [[differential Galois theory]], a domain studying which functions have [[antiderivative]]s of a prescribed form, which is able to give group-theoretic criteria when solutions of certain [[differential equation]]s are well-behaved.{{cref|u}} Geometric properties that remain stable under group actions are investigated in [[geometric invariant theory|(geometric)]] [[invariant theory]].<ref>{{Harvard citations|nb = yes|last1 = Mumford|last2 = Fogarty|last3 = Kirwan|year = 1994}}</ref> [[Image:Circle as Lie group2.svg|left|200px|thumb|The [[circle]] of center 0 and radius 1 in the [[complex plane]] is a Lie group under complex multiplication. Being a manifold means that every [[Neighbourhood (mathematics)|small piece]], such as the red arc in the figure, looks like a part of the [[real line]] (shown at the bottom).]] ===Lie groups=== {{Main|Lie group}} In many situations groups are endowed with an [[group object|additional structure]]. ''Lie groups'' (in honor of [[Sophus Lie]]) are groups which also have a compatible [[manifold]] structure, i.e. spaces [[diffeomorphism|looking locally like]] some [[euclidian space]] of the appropriate [[dimension]].<ref>{{Harvard citations|nb = yes|last = Warner|year = 1983}}</ref> Because of the manifold structure it is possible to consider continuous [[path (topology)|paths]] in the group. For this reason they are also referred to as continuous groups. Various Lie groups are important in physics. [[Rotation]], as well as [[translation (geometry)|translations]] in [[space]] and [[time]] are basic symmetries of the laws of [[mechanics]]. They can, for instance, be used to construct simple models – imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.{{cref|v}} Lie groups are also of more fundamental importance: [[Noether's theorem]] links continuous symmetries to [[conserved quantities]].<ref>{{Harvard citations|nb = yes|last = Goldstein|year = 1980}}</ref> The [[Poincaré group]] plays a pivotal role in [[special relativity]] and, by implication, for [[quantum field theories]].<ref>{{Harvard citations|nb = yes|last = Naber|year = 2003}}</ref> [[Local symmetry|Symmetries that vary with location]] are central to the modern description of physical interactions with the help of [[gauge theory]].<ref>See {{Harvard citations|txt = yes| last = Becchi|year = 1997}}</ref> [[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 ''x''-coordinate by factor 2.]] ====General linear group and matrix groups==== Many groups, especially Lie groups, can be described as groups of [[Matrix (mathematics)|matrices]] together with [[matrix multiplication]]. The ''[[general linear group]]'' {{nowrap begin}}''GL''(''n'', '''R'''){{nowrap end}} consists of all [[invertible matrix|invertible]] ''n''-by-''n'' matrices with [[real number|real]] entries.<ref>{{Harvard citations|nb = yes|last = Lay|year = 2003}}</ref> Being an [[open subset]] of the space of all ''n''-by-''n'' matrices, it is a Lie group.<ref>{{Harvard citations|nb = yes|last = Borel|year = 1991}}</ref> Its subgroups are referred to as ''[[linear group|matrix groups]]''. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the [[special orthogonal group]] ''SO''(''n''). It describes all possible rotations in ''n'' dimensions. Via [[Euler angles]], [[Rotation matrix|rotation matrices]] are used in [[computer graphics]].<ref>{{Harvard citations|nb = yes|last = Kuipers|year = 1999}}</ref> In chemical fields, such as [[crystallography]], [[space group]]s and their [[Character_theory#Character_tables|character tables]] are used to describe [[molecular symmetry|molecular symmetries]].<ref>{{Harvard citations|nb = yes|last1 = Conway|last2= Delgado Friedrichs|last3 = Huson|last4 = Thurston|year = 2001}}. See also {{Harvard citations|nb = yes|last = Bishop|year = 1993}}</ref> ===Representation theory=== {{Main|Representation theory}} ''Representation theory'' is both an application of the group concept and a major branch of group theory itself.<ref name="FultonHarris"/><ref>{{Harvard citations|nb = yes|last = Serre|year = 1977}}</ref> It studies the group by its [[group action|actions]] on other spaces. A broad class of [[group representation]]s are ''linear representations'', i.e. the group is acting on a [[vector space]], such as the 3-dimensional [[Euclidian space]] '''R'''<sup>3</sup>. A representation of ''G'' on an ''n''-[[dimension]]al real vector space is simply a group homomorphism :''&rho;'': ''G'' &rarr; ''GL''(''n'', '''R''') from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices which is accessible to explicit computations.{{cref|w}} Given a group action, this gives further means to study the object being acted on.{{cref|x}} On the other hand, it also yields information about the group. Group representations are particularly useful for finite groups, Lie groups, [[algebraic group]]s and (locally) [[compact group]]s.<ref name="FultonHarris"/><ref>{{Harvard citations|nb = yes|last = Rudin|year = 1990}}</ref> === Galois groups === {{Main|Galois group}} ''Galois groups'' are groups of substitutions of the solutions of [[polynomial equation]]s.<ref>{{Harvard citations|nb = yes|last = Robinson|year = 1996|loc = p. viii}}</ref><ref>{{Harvard citations|nb = yes|last = Artin|year = 1998}}</ref> For example, the solutions of the [[quadratic equation]] {{nowrap begin}}''ax''<sup>2</sup> + ''bx'' + ''c'' = 0{{nowrap end}} are given by :<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}.</math> Exchanging "+" and "&minus;" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for [[cubic equation|cubic]] and [[quartic equation]]s, but ''do not exist'' in general for [[quintic equation|degree 5]] and higher.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = Chapter VI}} (see in particular p. 273 for concrete examples).</ref> Abstract properties of Galois groups (in particular their [[solvable group|solvability]]) associated to polynomials give a criterion which polynomials do have all their solutions expressible by [[radical]]s, i.e. solutions expressible using solely addition, multiplication, and [[Nth root|roots]] similar to the formula above.<ref>{{Harvard citations|nb = yes|last = Lang|year = 2002|loc = p. 292 (Theorem VI.7.2)}}</ref> ==See also== <div style="-moz-column-count:3; column-count:3;"> * [[List of group theory topics]] * [[List of small groups]] </div> ==Notes== {{Div col|cols=3}}<font class=small> {{cnote|a|Since a two-sided inverse of ''a'' (i.e. an element ''c'' such that ''c''•''a'' and ''a''•''c'' both equal ''e'') is guaranteed to exist, the one-sided inverse ''b'' must be equal to ''c''.}} {{cnote|b|In the context of [[commutative algebra]], '''Z'''<sub>''n''</sub> is usually denoted {{nowrap|'''Z'''/''n'''''Z'''}} or '''Z'''/''n'', see for example {{Harvard citations|txt = yes|last = Eisenbud|year = 1995}}.}} {{cnote|c|However, a group is not determined by its lattice of subgroups. See {{Harvard citations|txt = yes|last = Suzuki|year = 1951}}.}} {{cnote|d|The fact that the group operation extends this [[canonical|canonically]] is an instance of a [[universal property]].}} {{cnote|e|For example, if ''G'' is finite, then the [[order of a group|size]] of any subgroup and any quotient group divides the size of ''G'', according to Lagrange's theorem.}} {{cnote|f|The word homomorphism derives from [[Ancient Greek|Greek]] ὁμός – the same and [[wikt:μορφή|μορφή]] – structure.}} {{cnote|g|The kernel of a homomorphism is always a normal subgroup. The converse statement is also true: any normal subgroup ''N'' is the kernel of the canonical map ''G''&nbsp;&rarr;&nbsp;''G''/''N'', see {{Harvard citations|txt = yes|last = Lang|year = 2005|loc = &#xa7;II.4, p. 45}}.}} {{cnote|h|The additive notation for elements of a cyclic group would be {{nowrap|''t'' • ''a'', ''t'' in '''Z'''}}.}} {{cnote|i|See the [[Seifert–van Kampen theorem]] for an example.}} {{cnote|j|An example is [[group cohomology]] of a group which equals the [[singular homology]] of its [[classifying space]].}} {{cnote|k|Elements which do have multiplicative inverses are called ''[[unit (ring theory)|units]]'', see {{Harvard citations|txt = yes|last = Lang|year = 2002|loc =&#xa7;II.1, p. 84}}.}} {{cnote|l|The transition from the integers to the rationals by adding fractions is generalized by the [[quotient field]].}} {{cnote|m|The same is true for any [[field (mathematics)|field]] ''F'' instead of '''Q'''. See {{Harvard citations|txt = yes|last = Lang|year = 2005|loc = &#xa7;III.1, p. 86}}.}} {{cnote|n|For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See {{Harvard citations|txt = yes|last = Lang|year = 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.}} {{cnote|o|The stated property is a possible definition of prime numbers. See [[prime element]].}} {{cnote|p|For example, the [[Diffie-Hellman]] protocol uses the [[discrete logarithm]].}} {{cnote|q|The groups of order at most 2000 are known. [[Up to]] isomorphism, there are about 49 billion. See {{Harvard citations|txt = yes|last1 = Besche|last2 = Eick|last3 = O'Brien|year = 2001}}.}} {{cnote|r|The gap between the classification of simple groups and the one of all groups lies in the [[extension problem]], a problem too hard to be solved in general. See {{Harvard citations|txt = yes|last = Aschbacher|year = 2004|loc = p. 737}}.}} {{cnote|s|Equivalently, a nontrivial group is simple if its only quotient groups are the trivial group and the group itself. See {{Harvard citations|txt = yes|last = Michler|year = 2006}}, {{Harvard citations|txt = yes|last = Carter|year = 1989}}.}} {{cnote|t|More rigorously, every group is the symmetry group of some [[graph (mathematics)|graph]], see {{Harvard citations|txt = yes|last = Frucht|year = 1939}}.}} {{cnote|u|More precisely, the [[monodromy]] action on the [[vector space]] of solutions of the differential equations is considered. See {{Harvard citations|txt = yes|last = Kuga|year=1993|loc = pp. 105&ndash;113}}.}} {{cnote|v|See [[Schwarzschild metric]] for an example where symmetry greatly reduces the complexity of physical systems.}} {{cnote|w|This was crucial to the classification of finite simple groups, for example. See {{Harvard citations|txt = yes|last = Aschbacher|year = 2004}}.}} {{cnote|x|See, for example, [[Schur's Lemma]] for the impact of a group action on [[simple module]]s. A more involved example is the [[Leray spectral sequence]] relating arithmetic information to geometric information via the action of the (absolute) Galois group.}} </font> {{Div col end}} ===Citations=== {{reflist|3}} ==References== {{Wikibooks|Abstract algebra/Groups}} ===General references=== * {{Citation | last1=Artin | first1=Michael | authorlink1=Michael Artin | title=Algebra | publisher=[[Prentice Hall]] | isbn=978-0-89871-510-1 | year=1991}}, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article. * {{Citation | last1=Devlin | first1=Keith | authorlink1=Keith Devlin | title=The Language of Mathematics: Making the Invisible Visible | publisher=Owl Books | isbn=978-0-8050-7254-9 | year=2000}}, Chapter 5 provides a layman-accessible explanation of groups. * {{Citation | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract algebra | publisher=Wiley | location=New York | edition=3rd| isbn=978-0-471-43334-7 | id={{MathSciNet | id = 2286236}} | year=2004}}. * {{Fulton-Harris}} * {{Citation | last1=Hall | first1=G. 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{{Citation | last1=Kleiner | first1=Israel | title=The evolution of group theory: a brief survey | url=http://www.jstor.org/sici?sici=0025-570X(198610)59%3A4%3C195%3ATEOGTA%3E2.0.CO%3B2-9 | id={{MathSciNet | id = 863090}} | year=1986 | journal=[[Mathematics Magazine]] | issn=0025-570X | volume=59 | issue=4 | pages=195–215}} * {{Citation | last1=Smith | first1=David Eugene | author1-link=David Eugene Smith | title=History of Modern Mathematics | url=http://www.gutenberg.org/etext/8746 | series=Mathematical Monographs, No. 1 | year=1906}} * {{Citation | last1=Wussing | first1=Hans | title=The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-45868-7 | year=2007}} ==External links== *{{mathworld|title=Group|urlname=Group}} *[http://planetmath.org/encyclopedia/Group.html Group] at [[PlanetMath]]. *[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Development_group_theory.html The development of group theory] at ''The MacTutor History of Mathematics archive''. 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