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BINARY OPERATION
Section 3.1
Definition of a Group
3.1.1Definition: Abinaryoperation ∗ onasetS isa
function∗∶ → fromtheset
ofallordered
pairsofelementsinS intoS.
Theoperation∗ issaidtobeassociative if
∗ ∗
∗ ∗ forall , , ∈ .
Anelement ∈ iscalledanidentity elementfor∗
if ∗
and ∗
forall ∈ .
If∗ hasanidentityelement ,and ∈ ,then ∈
issaidtobeaninverse for if ∗
and ∗
.
UNIQUENESS OF IDENTITY AND
INVERSE
3.1.1Proposition: Let∗ beanassociative
binaryoperationonasetS.
(a) Theoperation∗ hasatmostone
identityelement.
(b) If∗ hasanidentityelement,thenany
elementofS hasatmostoneinverse.
INVERSES
3.1.3Proposition: Let∗ beanassociative
binaryoperationonasetS.If∗ hasanidentity
elementand , ∈ haveinverses
and
,
respectively,thentheinverseof
existsandis
equalto ,andtheinverseof ∗ existsandis
equalto
∗
.
SOME NOTATION
• Normally,thebinaryoperationswework
withwillbenotatedmultiplicatively;thatis,
insteadof ∗ wewilljustwrite ⋅ or
simply .
• Inthecasewherethebinaryoperationalso
satisfiesthecommutativelaw ∗
∗ ,
wewilluseadditivenotation;thatis,instead
of ∗ wewillwrite
.
GROUP
3.1.4Definition:Let ,∗ denoteanonemptyset togetherwitha
binaryoperation∗ on .Thatis,thefollowingconditionmustbe
satisfied.
(i) Closure: Forall ,
elementof .
∈ ,theelement ∗
isawelldefined
Then iscalledagroup ifthefollowingpropertieshold.
(ii) Associativity: Forall , , ∈ ,wehave ∗
∗ ∗ .
∗
(iii) Identity: Thereexistsanidentityelement ∈ ,thatis,an
element ∈ suchthat ∗
and ∗
forall ∈ .
(iv) Inverses: Foreach ∈ thereexistsaninverse element
∈ ,thatis,anelement
∈ suchthat ∗
and
∗
.
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“COMPACT” DEFINITION OF A GROUP
3.1.4′Definition: AgroupisanonemptysetG
withanassociativebinaryoperation,suchthat
G containsanidentityelementforthe
operation,andeachelementofG hasan
inverseinG.
THE SYMMETRIC GROUP;
THE SYMMETRIC GROUP OF DEGREE n
Definition3.1.5: Thesetofallpermutationsofaset
S isdenotedbySym .Thesetofallpermutations
oftheset 1, 2, … , isdenotedby .
iscalledthesymmetric
ThegroupSym
group onS,and iscalledthesymmetricgroupof
degree .
Proposition2.1.6: IfS isanonemptyset,then
Sym
isagroupundertheoperationof
compositionoffunctions.
CANCELLATION PROPERTY FOR GROUPS
3.1.7Proposition(CancellationProperty
forGroups): Let beagroup,andlet
, , ∈ .
(a) If
,then
.
(b) If
,then
.
ABELIAN GROUPS
3.1.9Definition: Agroup issaidtobe
abelian if
forall , ∈ .
NOTES:
1. Theoperationismostoftendenoted
additively.
UNIQUENESS OF SOLUTIONS TO EQUATIONS AND GROUPS
3.1.8Proposition: If isagroupand
, ∈ ,theneachoftheequations
and
hasauniquesolution.
Conversely,if isanonemptysetwithan
associativebinaryoperationinwhichthe
equations
and
havesolutions
forall , ∈ ,then isagroup.
FINITE AND INFINITE GROUPS
3.1.10Definition: Agroup issaidtobea
finitegroup iftheset hasafinitenumberof
elements.Inthiscase,thenumberof
elementsiscalledtheorder of ,denotedby
| |.If isnotfinite,itissaidtobeaninfinite
group.
2. Theidentityelementiscalledthezero
element andisdenotedby0.Donot
confusethiswiththenumber0.
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THE GENERAL LINEAR GROUP
3.1.11Definition: Thesetofallinvertible
matriceswithentriesin iscalledthe
generallineargroupofdegreenoverthe
realnumbers,andisdenotedby
.
3.1.12Proposition: Theset
groupundermatrixmultiplication.
formsa
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