On Finite NeutroGroups of Type-NG

2020·0.21 MB·

by  A.A.A. Agboola

InternationalJournalofNeutrosophicScience(IJNS) Vol. 10,No. 2,PP.84-95,2020
On Finite NeutroGroups of Type-NG[1,2,4]
A.A.A.Agboola(cid:5)
DepartmentofMathematics,FederalUniversityofAgriculture,Abeokuta,Nigeria.
[email protected]
(cid:5)Incommemorationofthe60thbirthdayoftheauthor
Abstract
TheNeutroGroupsasalternativestotheclassicalgroupsareofdifferenttypeswithdifferentalgebraicprop-
erties. Inthispaper, wearegoingtostudyaclassofNeutroGroupsoftype-NG[1,2,4]. InthisclassofNeu-
troGroups,theclosurelaw,theaxiomofassociativityandexistenceofinversearetakingtobeeitherpartially
trueorpartiallyfalseforsomeelements;whiletheexistenceofidentityelementandaxiomofcommutativity
are taking to be totally true for all the elements. Several examples of NeutroGroups of type-NG[1,2,4] are
presented alongwith their basicproperties. It is shownthat Lagrange’s theorem holdsfor some NeutroSub-
groups of a NeutroGroup and failed to hold for some NeutroSubgroups of the same NeutroGroup. It is also
shownthattheunionoftwoNeutroSubgroupsofaNeutroGroupcanbeaNeutroSubgroupevenifoneisnot
containedintheother;andthattheintersectionoftwoNeutroSubgroupsmaynotbeaNeutroSubgroup. The
conceptsofNeutroQuotientGroupsandNeutroGroupHomomorphismsarepresentedandstudied. Itisshown
thatthefundamentalhomomorphismtheoremoftheclassicalgroupsisholdingintheclassofNeutroGroups
oftype-NG[1,2,4].
Keywords:NeutroGroup;AntiGroup;NeutroSubgroup;NeutroQuotientGroup;NeutroGroupHomomorphism.
1 Introduction and Preliminaries
In any classical algebraic structure (X,∗), the law of composition of the elements of X otherwise called a
binaryoperation∗iswelldefinedforalltheelementsofX thatis,x∗y ∈X ∀x,y ∈X;andaxiomslikeas-
sociativity,commutativity,distributivity,etc.definedonXwithrespectto∗aretotallytrueforalltheelements
ofX. ThecompositionsofelementsofX thiswayarerestrictiveanddonotreflectthereality. Itdoesnotgive
roomforcompositionsthatareeitherpartiallydefined,partiallyundefined(indeterminate),andpartiallyouter-
definedortotallyouterdefinedwithrespectto∗. Howeverinthedomainofknowledge,scienceandreality,the
lawofcompositionandaxiomsdefinedonX mayeitherbeonlypartiallydefined(partiallytrue),orpartially
undefined(partiallyfalse),ortotallyundefined(totallyfalse)withrespectto∗.Inanattempttomodelthereal-
itybyallowingthelawofcompositiononX tobeeitherpartiallydefined,partiallyundefined(indeterminate),
andpartiallyouterdefinedortotallyouterdefined,Smarandache[8]in2019introducedthenotionsofNetroDe-
finedandAntiDefinedlaws,aswellasthenotionsofNeutroAxiomandAntiAxiominspiredbyhisworkin[9],
whichhasgivenbirthtonewfieldsofresearchcalledNeutroStructuresandAntiStructures. Foranyclassical
algebraiclaworaxiomdefinedonX,therecorrespondneutrosophictriplets< Law,NeutroLaw,AntiLaw >
and < Axiom,NeutroAxiom,AntiAxiom > respectively. Smarandache in [7] studied NeutroAlgebras and
AntiAlgebrasandin[6],hestudiedPartialAlgebras,UniversalAlgebras,EffectAlgebrasandBoole’sPartial
AlgebrasandheshowedthatNeutroAlgebrasaregeneralizationofPartialAlgebras. RezaeiandSmarandache
[5]studiedNeutro-BE-algebrasandAnti-BE-algebrasandfundamentallytheyshowedthatanyclassicalalge-
braS withnoperations(lawsandaxioms)wheren ≥ 1willhave(2n −1)NeutroAlgebrasand(3n −2n)
AntiAlgebras. Agboolaetal. in[1]studiedNeutroAlgebrasandAntiAlgebrasviz-a-viztheclassicalnumber
systems N, Z, Q, R and C and in [2], he studied NeutroGroups by considering three NeutroAxioms (Neu-
troAssociativity,existenceofNeutroNeutralelementandexistenceofNeutroInverseelement). Inaddition,he
studied NeutroSubgroups, NeutroCyclicGroups, NeutroQuotientGroups and NeutroGroupHomomorphisms.
He showed that generally, Lagrange’s theorem and 1st isomorphism theorem of the classical groups do not
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holdintheclassofNeutroGroups. In[3],AgboolastudiedNeutroRingsbyconsideringthreeNeutroAxioms
(NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multiplication
overaddition)). HepresentedSeveralresultsandexamplesonNeutroRings,NeutroSubgrings,NeutroIdeals,
NeutroQuotientRingsandNeutroRingHomomorphisms. Heshowedthatthe1stisomorphismtheoremofthe
classicalringsholdsintheclassofNeutroRings. MotivatedandinspiredbytheworkofRezaeiandSmaran-
dache in [5], the work on NeutroGroups presented in [2] is revisited and the present work is devoted to the
study of a class of NeutroGroups of type-NG[1,2,4]. In this class of NeutroGroups, the closure law, the ax-
iom of associativity and existence of inverse are taking to be either partially true or partially false for some
elements;whiletheexistenceofidentityelementandaxiomofcommutativityaretakingtobetotallytruefor
all the elements. Several examples of NeutroGroups of type-NG[1,2,4] are presented along with their basic
properties. ItisshownthatLagrange’stheoremholdsforsomeNeutroSubgroupsofaNeutroGroupandfailed
toholdforsomeNeutroSubgroupsofthesameNeutroGroup. ItisalsoshownthattheunionoftwoNeutro-
SubgroupsofaNeutroGroupcanbeaNeutroSubgroupevenifoneisnotcontainedintheother;andthatthe
intersectionoftwoNeutroSubgroupsmaynotbeaNeutroSubgroup. TheconceptsofNeutroQuotientGroups
andNeutroGroupHomomorphismsarepresentedandstudied.Itisshownthatthefundamentalhomomorphism
theoremoftheclassicalgroupsisholdingintheclassofNeutroGroupsoftype-NG[1,2,4].
Definition1.1. [6]
(i) Aclassicaloperationisanoperationwelldefinedforalltheset’selements.
(ii) A NeutroOperation is an operation partially well defined, partially indeterminate, and partially outer
definedonthegivenset.
(iii) AnAntiOperationisanoperationthatisouterdefinedforallset’selements.
(iv) Aclassicallaw/axiomdefinedonanonemptysetisalaw/axiomthatistotallytrue(i.e. trueforallset’s
elements).
(v) ANeutroLaw/NeutroAxiom(orNeutrosophicLaw/NeutrosophicAxiom)definedonanonemptysetisa
law/axiomthatistrueforsomeset’selements[degreeoftruth(T)],indeterminateforotherset’selements
[degree of indeterminacy (I)], or false for the other set’s elements [degree of falsehood (F)], where
T,I,F ∈ [0,1],with(T,I,F) (cid:54)= (1,0,0)thatrepresentstheclassicalaxiom,and(T,I,F) (cid:54)= (0,0,1)
thatrepresentstheAntiAxiom.
(vi) AnAntiLaw/AntiAxiomdefinedonanonemptysetisalaw/axiomthatisfalseforallset’selements.
(vii) ANeutroAlgebraisanalgebrathathasatleastoneNeutroOperationoroneNeutroAxiom(axiomthat
istrueforsomeelements,indeterminateforotherelements,andfalseforotherelements),andnoAnti-
OperationorAntiAxiom.
(viii) AnAntiAlgebraisanalgebraendowedwithatleastoneAntiOperationoratleastoneAntiAxiom.
Theorem1.2. [5]LetUbeanonemptyfiniteorinfiniteuniverseofdiscourseandletS beafiniteorinfinite
subset of U. If n classical operations (laws and axioms) are defined on S where n ≥ 1, then there will be
(2n−1)NeutroAlgebrasand(3n−2n)AntiAlgebras.
2 Main Results
Definition2.1. [Classicalgroup][4]
LetGbeanonemptysetandlet∗ : G×G → GbeabinaryoperationonG. Thecouple(G,∗)iscalleda
classicalgroupifthefollowingconditionshold:
(G1) x∗y ∈G∀x,y ∈G[closurelaw].
(G2) x∗(y∗z)=(x∗y)∗z∀x,y,z ∈G[axiomofassociativity].
(G3) Thereexistse∈Gsuchthatx∗e=e∗x=x∀x∈G[axiomofexistenceofneutralelement].
(G4) Thereexistsy ∈Gsuchthatx∗y =y∗x=e∀x∈G[axiomofexistenceofinverseelement]where
eistheneutralelementofG.
Ifinaddition∀x,y ∈G,wehave
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(G5) x∗y =y∗x,then(G,∗)iscalledanabeliangroup.
Definition2.2. [NeutroSophicationofthelawandaxiomsoftheclassicalgroup]
(NG1) Thereexistatleastthreeduplets(x,y),(u,v),(p,q),∈ Gsuchthatx∗y ∈ G(degreeoftruthT)and
[u∗v =outer-defined/indeterminate(degreeofindeterminacyI)orp∗q (cid:54)∈G](degreeoffalsehoodF)
[NeutroClosureLaw].
(NG2) Thereexistatleastthreetriplets(x,y,z),(p,q,r),(u,v,w)∈Gsuchthatx∗(y∗z)=(x∗y)∗z(degree
oftruthT)and[[p∗(q∗r)]or[(p∗q)∗r]=outer-defined/indeterminate(degreeofindeterminacyI)or
u∗(v∗w)(cid:54)=(u∗v)∗w](degreeof,falsehoodF)[NeutroAxiomofassociativity(NeutroAssociativity)].
(NG3) Thereexistsanelemente ∈ Gsuchthatx∗e = e∗x = x(degreeoftruthT)and[[x∗e]or[e∗x] =
outer-defined/indeterminate(degreeofindeterminacyI)orx∗e (cid:54)= x (cid:54)= e∗x](degreeoffalsehoodF)
foratleastonex∈G[NeutroAxiomofexistenceofneutralelement(NeutroNeutralElement)].
(NG4) Thereexistsanelementu∈Gsuchthatx∗u=u∗x=e(degreeoftruthT)and[[x∗u]or[u∗x)]=
outer-defined/indeterminate(degreeofindeterminacyI)orx∗u (cid:54)= e (cid:54)= u∗x]foratleastonex ∈ G
(degreeoffalsehoodF)[NeutroAxiomofexistenceofinverseelement(NeutroInverseElement)]where
eisaNeutroNeutralElementinG.
(NG5) Thereexistatleastthreeduplets(x,y),(u,v),(p,q)∈Gsuchthatx∗y =y∗x(degreeoftruthT)and
[[u∗v]or[v∗u]=outer-defined/indeterminate(degreeofindeterminacyI)orp∗q (cid:54)=q∗p](degreeof
falsehoodF)[NeutroAxiomofcommutativity(NeutroCommutativity)].
Definition2.3. [AntiSophicationofthelawandaxiomsoftheclassicalgroup]
(AG1) Foralltheduplets(x,y)∈G,x∗y (cid:54)∈G[AntiClosureLaw].
(AG2) Forallthetriplets(x,y,z)∈G,x∗(y∗z)(cid:54)=(x∗y)∗z[AntiAxiomofassociativity(AntiAssociativity)].
(AG3) Theredoesnotexistanelemente∈Gsuchthatx∗e=e∗x=x∀x∈G[AntiAxiomofexistenceof
neutralelement(AntiNeutralElement)].
(AG4) Theredoesnotexistu ∈ Gsuchthatx∗u = u∗x = e∀x ∈ G[AntiAxiomofexistenceofinverse
element(AntiInverseElement)]whereeisanAntiNeutralElementinG.
(AG5) Foralltheduplets(x,y)∈G,x∗y (cid:54)=y∗x[AntiAxiomofcommutativity(AntiCommutativity)].
Definition2.4. [NeutroGroup]
ANeutroGroupNGisanalternativetotheclassicalgroupGthathasatleastoneNeutroLaworatleastone
of{NG1,NG2,NG3,NG4}withnoAntiLaworAntiAxiom.
Definition2.5. [AntiGroup]
An AntiGroup AG is an alternative to the classical group G that has at least one AntiLaw or at least one of
{AG1,AG2,AG3,AG4}.
Definition2.6. [NeutroAbelianGroup]
ANeutroAbelianGroupNGisanalternativetotheclassicalabeliangroupGthathasatleastoneNeutroLaw
oratleastoneof{NG1,NG2,NG3,NG4}andNG5withnoAntiLaworAntiAxiom.
Definition2.7. [AntiAbelianGroup]
AnAntiAbelianGroupAGisanalternativetotheclassicalabeliangroupGthathasatleastoneAntiLaworat
leastoneof{AG1,AG2,AG3,AG4}andAG5.
Proposition2.8. Let(G,∗)beafiniteorinfiniteclassicalnonabeliangroup. Then:
(i) thereare15typesofNeutroNonAbelianGroups,
(ii) thereare65typesofAntiNonAbelianGroups.
Proof. ItfollowsfromTheorem1.2.
Proposition2.9. Let(G,∗)beafiniteorinfiniteclassicalabeliangroup. Then:
(i) thereare31typesofNeutroAbelianGroups,
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(ii) thereare211typesofAntiAbelianGroups.
Proof. ItfollowsfromTheorem1.2.
Remark2.10. ItisevidentfromTheorem2.8andTheorem2.9thattherearemanytypesofNeutroGroupsand
NeutroAbelianGroups. ThetypeofNeutroGroupsstudiedbyAgboolain[2]isthatforwhichG1,G2,G3,G4
andG5areeitherpartiallytrueorpartiallyfalse.
Definition2.11. Let(NG,∗)beaNeutroGroup. NGissaidtobefiniteofordernifthecardinalityofNGis
nthatiso(NG)=n. Otherwise,NGiscalledaninfiniteNeutroGroupandwewriteo(NG)=∞.
Definition2.12. Let(AG,∗)beanAntiGroup. AGissaidtobefiniteofordernifthecardinalityofAGisn
thatiso(AG)=n. Otherwise,AGiscalledaninfiniteAntiGroupandwewriteo(AG)=∞.
Example 2.13. Let NG = N = {1,2,3,4··· ,}. Then (NG,.) is a finite NeutroGroup where (cid:48)(cid:48).(cid:48)(cid:48) is the
binaryoperationofordinarymultiplication.
Example 2.14. Let AG = Q∗ be the set of all irrational positive numbers and consider algebraic structure
+
(AG,∗) where ∗ is ordinary multiplication of numbers. It is clear that ∗ is a total AntiLaw defined on AG.
Thebinaryoperation∗istotallyassociativeforallthetriplets(x,y,z)withx,y,z ∈AG. Thereisnoneutral
element(s) for all the elements AG and hence no element of AG has an inverse. Finally, the operation ∗
is commutative for all the duplets (x,y) with x,y ∈ AG. Hence by Definition 2.5, (AG,∗) is an infinite
AntiGroup.
Example2.15. LetU={a,b,c,d,e,f}beauniverseofdiscourseandletAG={a,b,c,}beasubsetofU.
Let∗beabinaryoperationdefinedonAGasshownintheCayleytablebelow:
∗ a b c
a d c b
.
b c e a
c b a f
Itisclearfromthetablethatexceptforthecompositionsa∗a=d,b∗b=e,c∗c=f thatareouter-defined
withthedegreeoffalsity33%,therestcompositionsareinner-definedwith66.7%degreeoftruth. Thisshows
that G1 is partially true and partially false so that ∗ is a NeutroLaw. Also, G2 is partial true and partially
false. There are 33 = 27 possible triplets out of which only 6 can verify associativity of ∗. Hence degree
of associativity of ∗ is 22.2% while the degree of non-associativity is 77.8% so that ∗ is NeutroAssociative.
However,G3andG4aretotallyfalseforalltheelementsofAGwhichshowsthatAG3andAG4aresatisfied.
Lastly,G5ispartiallytruewiththedegreeoftruth50%andpartiallyfalsewith50%degreeoffalsitywhich
showsthat∗isNeutroCommutative. HencebyDefinition2.5,(AG,∗)isafiniteAntiGroup.
3 Certain Types of NeutroGroups
Inthissection,wearegoingtostudycertaintypesofNeutroGroups(NG,∗).TheNeutroGroupswillbenamed
accordingtowhichofNG1−NG5is(are)satisfied. Inthesequel,x∗ywillbewrittenasxy∀x,y ∈NG.
Example3.1. LetU={a,b,c,d,e,f}beauniverseofdiscourseandletNG={e,a,b,c}beasubsetofU.
Let∗beabinaryoperationdefinedonNGasshownintheCayleytablebelow:
∗ e a b c
e e a b c
a a b a b .
b b c f c
c c d c e
ItisclearfromthetablethatG1,G2,G3,G4andG5arepartiallytrueandpartiallyfalsewithrespectto∗as
shownbelow:
(i) NeutroClosureLaw (NG1): Except for the compositions b∗b = f,c∗a = d which are false with
12.5%degreeoffalsity,allothercompositionsaretruewith87.5%degreeoftruth.
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(ii) NeutroAssociativity(NG2):
a∗(b∗c) = (a∗b)∗c=b.
a∗(a∗a) = a, but,(a∗a)∗a=c(cid:54)=a.
(iii) NeutroNeutralElement(NG3):
N = N =N =e but
e a b
N = e or b.
c
(iv) NeutroInverseElement(NG4):
I = e,
e
I doesnotexist,
a
I doesnotexist,
b
I = e.
c
(v) NeutroCommutativity(NG5):
b∗c = c∗b=c.
a∗b = a but b∗a=c(cid:54)=a.
We have just shown that (NG,∗) is a finite NeutroAbelianGroup. This is an example of the class of Neu-
troGroupsstudiedbyAgboolain[2]. ThisclassofNeutroGroupsarereferredtoasoftype-NG[1,2,3,4,5].
Example3.2. LetU={a,b,c,d,e,f}beauniverseofdiscourseandletNG={a,b,c,e}beasubsetofU.
Let∗beabinaryoperationdefinedonNGasshownintheCayleytablebelow:
∗ a b c e
a e c f a
b c e d b .
c d a e c
e a b c e
ItisclearfromthetablethatG3andG4aretotallytrueforalltheelementsofNG. However,G1,G2andG5
arepartiallytrueandpartiallyfalsewithrespectto∗asshownbelow:
(i) NeutroClosureLaw(NG1):Exceptforthecompositionsa∗c=f,b∗c=d,c∗a=dwhichareouter-
definedwith18.75%degreeoffalsity, allothercompositionsareinner-definedwith81.25%degreeof
truth.
(ii) NeutroAssociativity(NG2):
c∗(b∗b) = (c∗b)∗b=c.
a∗(b∗c) = a∗d=outer-defined,(a∗b)∗c=e.
(iii) NeutroCommutativity(NG5):
a∗b = b∗a=c.
a∗c = f but c∗a=d(cid:54)=f.
Wehavejustshownthat(NG,∗)isafiniteNeutroAbelianGroupoftype-NG[1,2,5].
Example3.3. LetNG=Zandlet∗beabinaryoperationdefinedonNGby
x∗y =x+xy x,y ∈Z.
ItisclearthatonlyG1istotallytrueforallx,y ∈NGbutG2,G3,G4andG5arepartiallytrueandpartially
falsewithrespectto∗asshownbelow:
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(i) NeutroAssociativity(NG2):
x∗(y∗z) = x+xy+xyz
(x∗y)∗z = x+xy+xz+xyz byequating,wehave
x+xy+xyz = x+xy+xz+xyz
⇒ xz = 0 fromwhichweobtain
x = 0 or z =0.
Thisshowsthatonlythetriplets(0,y,0),(0,y,z),(x,y,0)canverifyassociativityof∗.
(ii) NeutroNeutralElement(NG3):
Itisclearthatonlytheelement0∈NGhas0asitsneutralelementandnoneutral(s)forotherelements
ofNG.
(iii) NeutroInverseElement(NG4):
Again, onlytheelement0 ∈ NGhas0astheinverseelementandnoinverse(s)forotherelementsof
NG.
(iv) NeutroCommutativity(NG5):
Only the duplet (0,0) can verify the commutativity of ∗ and not any other duplet(s) (x,y). Hence,
(NG,∗)isaninfiniteNeutroAbelianGroupoftype-NG[2,3,4,5].
Definition3.4. Let(NG,∗)beaNeutroGroup. AnonemptysubsetNH ofNGiscalledaNeutroSubgroup
of NG if (NH,∗) is also a NeutroGroup of the same type as NG. If (NH,∗) is a NeutroGroup of a type
differentfromthatofNG,thenNH willbecalledaQuasiNeutroSubgroupofNG.
Example3.5. Let(NG,∗)betheNeutroGroupofExample3.2andletNH ={a,c,e}andNH ={b,c,e}
1 2
betwosubsetsofNG. Let∗bedefinedonNH andNH asshownintheCayleytablesbelow:
1 2
∗ a c e ∗ b c e
a e f a b e d b
NH : NH :
1 c d e c 2 c a e c
e a c e e b c e
It can easily be shown that (NH ,∗) and (NH ,∗) are NeutroGroups of type-NG[1,2,5] and therefore
1 2
NH andNH areNeutroSubgroupsofNG. Wenotethato(NG) = 4,o(NH ) = 3 = o(NH ). Since3
1 2 1 2
doesnotdivide4,itfollowsthatLagranges’theoremdoesnothold. Nowconsiderthefollowing:
NH ∪NH = {a,b,c,e}=NG.
1 2
NH ∩NH = {c,e}.
1 2
TheseshowthatNH ∪NH isaNeutroSubgroupofNGbutNH ∩NH isnotaNeutrosubgroupofNG.
1 2 1 2
ItishoweverobservedthatNH ∩NH isagroupascanbeseenintheCayleytablebelow:
1 2
∗ c e
NH ∩NH : c e c .
1 2
e c e
Definition3.6. Let(NG,∗)beaNeutroGroupandleta∈NGbeafixedelement.
(i) ThecenterofNGdenotedbyZ(NG)isasetdefinedby
Z(NG)={x∈NG:xg =gx foratleastone g ∈NG}.
(ii) Thecentralizerofa∈GdenotedbyNC isasetdefinedby
a
NC ={g ∈NG:ga=ag}.
a
Example3.7. Let(NG,∗)betheNeutroGroupofExample3.2. Then:
(i) Z(NG)={a,b,c,e}=NG. ThisshowsthatZ(NG)isaNeutroSubgroupofNG.
(ii) NC = {a,b,e}, NC = {a,b,e}, NC = {c,e} and NC = {a,b,c,e}. We have that NC and
a b c e a
NC arenotNeutroSubgroupsofNG,NC isagroupandNC isaNeutroSubgroupofNG.
b e c
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4 Characterization of Finite NeutroGroups of type-NG[1,2,4]
In this section, we are going to study finite NeutroGroups of type-NG[1,2,4] that is NeutroGroups (NG,∗)
whereG3andG5aretotallytrueforalltheelementsofNGandwhereG1, G2andG4areeitherpartially
trueorpartiallyfalseforsomeelementsofNG.
Example4.1. LetNG={1,2,3,4}⊆Z andlet∗beabinaryoperationonNGdefinedas
5
x∗y =x+y+4 ∀x,y ∈NG.
Then(NG,∗)isafiniteNeutroGroupoftype-NG[1,2,4]ascanbeseenintheCayleytable
∗ 1 2 3 4
1 1 2 3 4
2 2 3 4 0 .
3 3 4 0 1
4 4 0 1 2
Example 4.2. Let NG = {1,2,3} ⊆ Z and let • be a binary operation defined on NG as shown in the
4
Cayleytable
(cid:63) 1 2 3
1 1 2 3
.
2 2 0 2
3 3 2 1
Then(NG,•)isafiniteNeutroGroupoftype-NG[1,2,4].
Example4.3. LetNK ={1,3,5}⊆Z andlet◦beabinaryoperationdefineonNKasshownintheCayley
8
table
◦ 1 3 5
1 1 3 5
.
3 3 1 7
5 5 7 1
Then(NK,◦)isafiniteNeutroGroupoftype-NG[1,2,4].
Example4.4. LetNG = {1,2,3,4,5} ⊆ Z andlet∗beabinaryoperationdefineonNGasshowninthe
10
Cayleytable
∗ 1 2 3 4 5
1 1 2 3 4 5
2 2 4 6 8 0
.
3 3 6 9 2 5
4 4 8 2 6 0
5 5 0 5 0 5
Then(NG,∗)isafiniteNeutroGroupoftype-NG[1,2,4].
Example4.5. Let(NG,•)and(NK,◦)beNeutroGroupsofExamples4.2and4.3respectively. Then
NG×NG = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)},
NK×NK = {(1,1),(1,3),(1,5),(3,1),(3,3),(3,5),(5,1),(5,3),(5,5)},
NG×NK = {(1,1),(1,3),(1,5),(2,1),(2,3),(2,5),(3,1),(3,3),(3,5)}.
Itcaneasilybeshownthat(NG×NG,•),(NK×NK,◦)and(NG×NK,(cid:50))areNeutroGroupsoftype-
NG[1,2,4].
Proposition 4.6. Let (NG,•) and (NK,◦) be any two NeutroGroups of the same type-NG[1,2,4]. Then
(NG×NG,•),(NK×NK,◦)and(NG×NK,(cid:50))areNeutroGroupsoftype-NG[1,2,4].
Proof. Theproofiseasyandsoomitted.
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Proposition4.7. Let(NG,∗)beafiniteNeutroGroupoftype-NG[1,2,4]. Theclassicallawsofindicesdonot
hold.
Proof. Since∗isaNeutroLawand∗isNeutroAssociativeoverNG,theresultfollows.
Corollary 4.8. No NeutroGroup (NG,∗) of type-NG[1,2,4] can be cyclic that is, generated by an element
x∈NG.
Proposition4.9. Let(NG,∗)beafiniteNeutroGroupoftype-NG[1,2,4]. Ifxandy areinvertibleelements
ofNG,then
(i) (cid:0)x−1(cid:1)−1 =x.
(ii) (xy)−1 =x−1y−1.
Proof. Obvious.
Example4.10. Let(NG,∗)betheNeutroGroupofExample4.1andconsiderthefollowingsubsetsofNG.
NH ={1,2},NH ={1,3},NH ={1,4},NH ={1,2,3},NH ={1,2,4},NH ={1,3,4}.
1 2 3 4 5 6
Itcanbeshownthat(NH ,∗),i=1,2,3,4,5,6areNeutroSubgroupsofNG. Nextconsiderthefollowing:
i
NH ∪NH = NH ∪NH ={1,2,3} [NeutroSubgroupsofNG].
1 2 1 4
NH ∪NH = NH ∪NH =NH ∪NH ={1,2,4} [NeutroSubgroupsofNG].
1 3 1 5 3 5
NH ∪NH = NH ∪NH ={1,3,4} [NeutroSubgroupsofNG].
2 3 2 6
NH ∪NH = NH ∪NH ={1,2,3,4} [trivialNeutroSubgroupsofNG].
3 4 5 6
NH ∩NH = NH ∩NH =NH ∩NH ={1,2} [NeutroSubgroupsofNG].
1 4 1 5 4 5
NH ∩NH = NH ∩NH =NH ∩NH ={1,3} NeutroSubgroupsofNG].
2 4 2 6 4 6
NH ∩NH = NH ∩NH ={1,4} [NeutroSubgroupsofNG].
3 5 3 6
NH ∩NH = NH ∩NH ={1} [notNeutroSubgroupsofNG].
1 2 2 3
Example4.11. Let(NG,∗)betheNeutroGroupofExample4.4andconsiderthefollowingsubsetsofNG.
NH ={1,2,4,5},NH ={1,2,3,5}.
1 2
Itcanbeshownthat(NH ,∗),i=1,2areNeutroSubgroupsofNG. Nextconsiderthefollowing:
i
NH ∪NH = ={1,2,5} [aNeutroSubgroupofNG].
1 2
NH ∩NH = {1,2,3,4,5} [atrivialNeutroSubgroupofNG].
1 2
Remark4.12. Examples4.10and4.11haveshownthatintheNeutroGroupsoftype-NG[1,2,4],wecanhave
thefollowing:
(i) Lagrange’stheoremmayholdforsomeNeutroSubgroupsoftheNeutroGroupsandfailtoholdforsome
NeutroSubgroups.
(ii) The union of two NeutroSubgroups of the NeutroGroups can be NeutroSubgroups even if one is not
containedintheother.
(iii) TheintersectionoftwoNeutroSubgroupsoftheNeutroGroupscanbeNeutroSubgroups.
Definition4.13. LetNH beaNeutroSubgroupoftheNeutroGroup(NG,∗)andletx∈NG.
(i) xNH theleftcosetofNH inNGisdefinedby
xNH ={xh:h∈NH}.
(ii) ThenumberofdistinctleftcosetsofNH inNGiscalledtheindexofNH inNGdenotedby[NG :
NH].
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(iii) ThesetofalldistinctleftcosetsofNH inNGdenotedbyNG/NH isdefinedby
NG/NH ={xNH :x∈NG}.
Example4.14. Let(NG,∗)betheNeutroGroupofExample4.1andlet(NH ,∗),i = 1,2,3,4,5,6bethe
i
NeutroSubgroupsofExample4.10. TheleftcosetsofNH arecomputedasfollows.
i
1NH = {1,2},2NH ={2,3},3NH ={3,4},4NH ={0,4}.
1 1 1 1
1NH = {1,3},2NH ={2,4},3NH ={0,3},4NH ={1,4}.
2 2 2 2
1NH = {1,4},2NH ={0,2},3NH ={1,3},4NH ={2,4}.
3 3 3 3
1NH = {1,2,3},2NH ={2,3,4},3NH ={0,3,4},4NH ={0,1,4}.
4 4 4 4
1NH = {1,2,4},2NH ={0,2,3},3NH ={1,3,4},4NH ={0,2,4}.
5 5 5 5
1NH = {1,3,4},2NH ={0,2,4},3NH ={0,1,3},4NH ={1,2,4}.
6 6 6 6
∴ NG/NH = {1NH ,2NH ,3NH ,4NH },i=1,2,3,4,5,6.
i i i i i
Example4.15. Let(NG,∗)betheNeutroGroupofExample4.4andlet(NH ,∗),i = 1,2betheNeutro-
i
SubgroupsofExample4.11. TheleftcosetsofNH arecomputedasfollows.
i
1NH = {1,2,4,5},2NH ={0,2,4,8},3NH ={2,3,5,6},4NH ={0,4,6,8},5NH ={0,5}.
1 1 1 1 1
1NH = {1,2,3,5},2NH ={0,2,4,6},3NH ={3,5,6,9},4NH ={0,2,4,8},5NH ={0,5}.
2 2 2 2 1
∴ NG/NH = {1NH ,2NH ,3NH ,4NH ,5NH },i=1,2.
i i i i i i
Lemma4.16. LetNH beaNeutroSubgroupoftheNeutroGroup(NG,∗)oftype-NG[1,2,4]andletx∈NG.
Then,xNH =NH ifandonlyifx=ewhereeistheidentityelementinNG.
Proof. Obvious.
Remark4.17. Examples4.14and4.15haveshownthatintheNeutroGroupsoftype-NG[1,2,4]distinctleft
cosetsofNeutroSubgroupsintheNeutroGroupsdonotnecessarilypartitiontheNeutroGroups.
LetNHbeaNeutroSubgroupofaNeutroGroup(NG,∗)oftype-NG[1,2,4]andletNG/NHbethesetof
distinctleftcosetsofNH inNG. ForxNH,yNH ∈NG/NH withx,y ∈NG,let(cid:12)beabinaryoperation
definedonNG/NH by
xNH (cid:12)yNH =xyNH ∀x,y ∈NG.
Wewanttoinvestigateifthecouple(NG/NH,(cid:12))isaNeutroGroupoftype-NG[1,2,4]usingthefollowing
examples.
Example4.18. LetNG/NH = {1NH ,2NH ,3NH ,4NH },i = 1,2,3,4,5,6beasgiveninExample
i i i i i
4.14. Fori=1,wehave
(cid:12) 1NH 2NH 3NH 4NH
1 1 1 1
1NH 1NH 2NH 3NH 4NH
1 1 1 1 1
2NH 2NH 3NH 4NH 0NH .
1 1 1 1 1
3NH 3NH 4NH 0NH 1NH
1 1 1 1 1
4NH 4NH 0NH 1NH 2NH
1 1 1 1 1
ItisclearfromtheCayleytablethat(NG/NH ,(cid:12))isaNeutroGroupoftype-NG[1,2,4]with1NH asthe
1 1
identityelement. Thisisalsotruefori=2,3,4,5,6.
Example 4.19. Let NG/NH = {1NH ,2NH ,3NH ,4NH ,5NH },i = 1,2 be as given in Example
i i i i i i
4.15. Fori=1,wehave
(cid:12) 1NH 2NH 3NH 4NH 5NH
1 1 1 1 1
1NH 1NH 2NH 3NH 4NH 5NH
1 1 1 1 1 1
2NH 2NH 4NH 6NH 8NH 0NH
1 1 1 1 1 1 .
3NH 3NH 6NH 9NH 2NH 5NH
1 1 1 1 1 1
4NH 4NH 8NH 2NH 6NH 0NH
1 1 1 1 1 1
5NH 5NH 0NH 5NH 0NH 5NH
1 1 1 1 1 1
ItisevidentfromtheCayleytablethat(NG/NH ,(cid:12))isaNeutroGroupoftype-NG[1,2,4]with1NH asthe
1 1
identityelement.. Thisisalsotruefori=2.
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Remark4.20. ItisevidentfromExamples4.18and4.19thatifNH isaNeutroSubgroupoftheNeutroGroup
oftype-NG[1,2,4],thenNG/NH thesetofdistinctleftcosetscanbemadeaNeutroGroupoftype-NG[1,2,4]
bydefiningappropriatebinaryoperation(cid:12)onNG/NH. TheNeutroGroupNG/NH iscalledtheQuotient-
NeutroGroupofNGfactoredbyNH.
Definition 4.21. Let (NG,∗) and (NH,◦) be any two NeutroGroups of type-NG[1,2,4]. The mapping φ :
NG→NH iscalledaNeutroGroupHomomorphismifφpreservesthebinaryoperations∗and◦thatisiffor
atleastaduplet(x,y)∈G,wehave
φ(x∗y)=φ(x)◦φ(y).
TheKernelofφdenotedbyKerφisdefinedby
Kerφ={x:φ(x)=e }
NH
wheree istheidentityelementinNH.
NH
TheImageofφdenotedbyImφisdefinedby
Imφ={y ∈H :y =φ(x) forsome h∈NH}.
IfinadditionφisaNeutroBijection,thenφiscalledaNeutroGroupIsomorphismandwewriteNG ∼= NH.
NeutroGroupEpimorphism,NeutroGroupMonomorphism,NeutroGroupEndomorphism,andNeutroGroupAu-
tomorphismaresimilarlydefined.
Example4.22. Let(NG,•)betheNeutroGroupofExample4.2andletφ:NG×NG→NGbeaprojection
givenby
ψ(x,y)=y ∀x,y ∈NG.
Then
φ(1,1)=φ(2,1)=φ(3,1)=1,φ(1,2)=φ(2,2)=φ(3,2)=2,φ(1,3)=φ(2,3)=φ(3,3)=3.
Since φ((2,1)(3,3)) = φ(2,3) = 3 and φ(2,1)φ(3,3) = 1•3 = 3 but φ((2,2)(2,3)) = φ(0,2) =? and
φ(2,2)φ(2,3) = 2•3 = 2,itfollowsthatφisaNeutroGroupHomomorphism. Imφ = {1,2,3} = NGand
Kerφ={(1,1),(2,1),(3,1)}. TheKerφisaNeutroSubgroupofNG×NGascanbeseeninthefollowing
Cayleytable
• (1,1) (2,1) (3,1)
(1,1) (1,1) (2,1) (3,1)
.
(2,1) (2,1) (0,1) (2,1)
(3,1) (3,1) (2,1) (1,1)
Itisevidentfromthetablethat(Kerφ,•)isaNeutroGroupoftype-NG[1,2,4]andsinceKerφ⊆NG×NG,
itfollowsthatkerφisaNeuroSubgroup.
Example 4.23. Let (NK,◦) be the NeutroGroup of Example 4.3 and let ψ : NK × NK → NK be a
projectiongivenby
ψ(x,y)=x ∀x,y ∈NK.
Then
ψ(1,1)=ψ(1,3)=ψ(1,5)=1,ψ(3,1)=ψ(3,3)=ψ(3,5)=3,ψ(5,1)=ψ(5,3)=ψ(5,5)=5.
Sinceψ((1,1)(1,3)) = ψ(1,3) = 1andψ(1,1)ψ(1,3) = 1◦1 = 1butψ((1,5)(5,3)) = ψ(5,7) =?and
ψ(1,5)ψ(5,3) = 1◦5 = 5, it follows that ψ is a NeutroGroupHomomorphism. Imψ = {1,3,5} = NK
and Kerψ = {(1,1),(1,3),(1,5)}. The Kerψ is a NeutroSubgroup of NK ×NK as can be seen in the
followingCayleytable
◦ (1,1) (1,3) (1,5)
(1,1) (1,1) (1,3) (1,5)
.
(1,3) (1,3) (1,1) (1,7)
(1,5) (1,5) (1,7) (1,1)
Itisevidentfromthetablethat(Kerψ,◦)isaNeutroGroupoftype-NG[1,2,4]andsinceKerψ ⊆NK×NK,
itfollowsthatkerψisaNeuroSubgroup.
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