Group (mathematics)

In mathematics,group isset, together withbinary operation satisfying certain axioms, detailed below. The branchmathematics which studies groupscalled group theory.

The historical origingroup theory goes back toworksEvariste Galois (1830), concerningproblemwhen an algebraic equationsoluble by radicals.

A great many ofobjects investigatedmathematics turn outbe groups, including familiar number systems, such asintegers, rational, real,complex numbers under addition, non-zero rational, real,complex numbers under multiplication, non-singular matricies under multiplication, invertable functions under composition,so on. Group Theory allows forpropertiesthese systemsmany othersbe investigated inmore general setting,its resultswidely applicable. Group theoryalsorich sourcetheoremsits own right. Groups underlieother algebraic structures such as fieldssvector spaces andalso important toolsstudying symmetryall its forms. For these reasons, group theoryconsideredbe an important areamodern mathematics, and has many applicationsmathematical physics (for example,particle theory)

Tablecontents

1 History 2 Basic definitions 3 Notationgroups 4 Some elementary examplesnonexamples 4.1 An abelian group:integers under addition 4.2 Notgroup:integers under multiplication 4.3 An abelian group:nonzero rational numbers under multiplication 4.4 A finite nonabelian group: permutations ofset 4.5 Further examples 5 Simple theorems 6 Constructing new groups from given ones 7 Related topics

History

See Group theory.

Basic definitions

A group (G,*)defined asset G together withbinary operation *: G × G → G. We write "a * b" forresultapplyingoperation * totwo elements abG. To havegroup, * must satisfyfollowing axioms:

• Associativity: For all a, bcG, (a * b) * c = a * (b * c).
• Identity element: There is an element eG such thatall aG, e * a = a = a * e.
• Inverse element: For all aG, therean element bG such that a * b = e = b * a, where e isidentity element fromprevious axiom.

You will often also seeaxiom

• Closure: For all abG, a * b belongsG.

The way thatdefinition abovephrased, this axiom isn't necessary, since binary operationsalready requiredsatisfy closure. When determining if * isgroup operation, however, itnonetheless necessaryverify that * satisfies closure; thispartverifying that itin factbinary operation.

It should be noted that thereno requirement ingroup that a * b = b * a (commutativity). A groupwhich this equation holdsall abG,called abelian (aftermathematican Niels Abel). Groups lacking this propertycalled non-abelian.

The order ofgroup G, denoted by |G| or o(G), isnumberelements ofset G. A groupcalled finite ifhas finitely many elements, thatifset G isfinite set.

Note that we often refer togroup (G,*) as simply "G", leavingoperation * unmentioned. Butbe perfectly precise, different operations onsame set define different groups.

Notationgroups

Usuallyoperation, whateverreally is,thoughtas an analoguemultiplication, andgroup operationstherefore written multiplicatively. That is:

• We write "a · b" or even "ab"a * bcall itproductab;
• We write "1" foridentity elementcall itunit element;
• We write "a⁻¹" forinverseacall itreciprocala.

However sometimesgroupthoughtas analogousadditionwritten additively:

• We write "a + b"a * bcall itsumab;
• We write "0" foridentity elementcall itzero element;
• We write "−a" forinverseacall itoppositea.

Usually, only abelian groupswritten additively.

When being noncommital, one can usenotation (with "*")terminology that was introduced indefinition, usingnotation a⁻¹ forinversea.

If S issubsetG,x an elementG thenmultiplicative notation, xS issetall products {xs}sS; similarlynotation Sx = {sx : sS};for two subsets STG, we write ST{st :all sS, tT}. In additive notation, we write x + S, S + x,S + T forrespective sets.

Some elementary examplesnonexamples

An abelian group:integers under addition

A group that weintroducedin elementary school isintegers under addition. For this example, let Z besetintegers, {...,−4,−3,−2,−1,0,1,2,3,4,...},letsymbol "+" indicateoperationaddition. Then (Z,+) isgroup (written additively).

Proof:

• If abintegers then a + ban integer. (Closure; + really isbinary operation)
• If a, b,cintegers, then (a + b) + c = a + (b + c). (Associativity)
• 0an integerfor any integer a, 0 + a = a = a + 0. (Identity element)
• If aan integer, then therean integer b := −a, such that a + b = 0 = b + a. (Inverse element)

This groupalso abelian: a + b = b + a.

The integersboth additionmultiplication together formmore complicated algebraic structure ofring. In fact,elementsany ring form an abelian group under addition, calledadditive group ofring.

Notgroup:integers under multiplication

Onother hand, if we consideroperationmultiplication, denoted by "·", then (Z,·)notgroup:

• If abintegers then a · ban integer. (Closure; · really isbinary operation)
• If a, b,cintegers, then (a · b) · c = a · (b · c). (Associativity)
• 1an integerfor any integer a, 1 · a = a = a · 1. (Identity element)
• But, if aan integer, therenot necessarily an integer b such that a · b = 1 = b · a. There may berational number b like that, but not an integer. (Inverse element fails)

So we see that not every element(Z,·) has an inversetherefore, (Z,·)notgroup. The most we can saythatismonoid.

An abelian group:nonzero rational numbers under multiplication

Considersetrational numbers Q, that issetnumbers a/b such that abintegersbnonzero, andoperation multiplication, denoted by "·". Sincerational number 0 does not havemultiplicative inverse, (Q,·), like (Z,·),notgroup.

However, if we instead useset Q \\ {0} insteadQ, thatinclude every rational number except zero, then (Q \\ {0},·) does form an abelian group (written multiplicatively). The inversea/bb/a, andother group axiomssimplecheck. We don't lose closure by removing zero, becauseproducttwo nonzero rationalsnever zero.

Just asintegers formring, sorational numbers formalgebraic structure offield. In fact,nonzero elementsany given field formgroup under multiplication, calledmultiplicative group offield.

A finite nonabelian group: permutations ofset

Formore abstract example, consider three colored blocks (red, green,blue), initially placed inorder RGB. Let a beaction "swapfirst block andsecond block",let b beaction "swapsecond block andthird block".

In multiplicative form, we traditionally write xy forcombined action "first do y, then do x"; so that ab isaction RGB → RBG → BRG, i.e., "takelast blockmovetofront". If we write e"leaveblocks asare" (the identity action), then we can writesix permutations ofsetthree blocks asfollowing actions:

• e : RGB → RGB
• a : RGB → GRB
• b : RGB → RBG
• ab : RGB → BRG
• ba : RGB → GBR
• aba : RGB → BGR

Note thataction aa haseffect RGB → GRB → RGB, leavingblocks aswere; so we can write aa = e. Similarly,

• bb = e,
• (aba)(aba) = e, and
• (ab)(ba) = (ba)(ab) = e;

so each ofabove actions has an inverse.

By inspection, we can also determine associativityclosure; noteexample that

• (ab)a = a(ba) = aba, and
• (ba)b = b(ab) = aba.

This groupcalledsymmetric group on 3 letters, or S₃. It has order 6 (or 3 factorial),is non-abelian (since,example, ab ≠ ba). Since S₃built up frombasic actions ab, we say thatset {a,b} generates it.

Every group can be expressedtermspermutation groups like S₃; this resultCayley's theoremis studied as part ofsubjectgroup actions.

Further examples

For some further examplesgroups fromvarietyapplications, see ExamplesgroupsListsmall groups.

Simple theorems

• A group has exactly one identity element.
• Every element has exactly one inverse.
• You can perform divisiongroups; that is, given elements ab ofgroup G, thereexactly one solution xG toequation x * a = bexactly one solution yG toequation a * y = b.
• The expression "a₁ * a₂ * ··· * a_n"unambiguous, becauseresult will besame no matter where we place parentheses.
• The inverse ofproduct isproduct ofinverses inopposite order: (a * b)⁻¹ = b⁻¹ * a⁻¹.

Theseother basic facts that holdall individual groups formfieldelementary group theory.

Constructing new groups from given ones

1. Ifsubset H ofgroup (G,*) together withoperation * restricted on Hitselfgroup, then itcalledsubgroup(G,*).
2. The direct sumtwo groups (G,*)(H,•) isset G×H together withoperation (g₁,h₁)(g₂,h₂) = (g₁*g₂,h₁•h₂).
3. Givengroup G andnormal subgroup N,quotient group issetcosetsG/N together withoperation (gN)(hN)=ghN.

Related topics

See Glossarygroup theorymore definitionsgroup theory.

See elementary group theory forlistelementary theoremsgroup theory.

See Listgroup theory topics forlistall group theory topics coveredWikipedia.