On NeutroNilpotentGroups

Neutrosophic Sets and Systems,Vol.38,2020

OnNeutroNilpotentGroups

ElaheMohammadzadeh1 andAkbarRezaei2, 1,2 DepartmentofMathematics,PayameNoorUniversity,P.O.Box19395-3697,Tehran,Iran e-mails: 1 mohamadzadeh36@gmail.com, 2 rezaei@pnu.ac.ir

∗ Correspondence:rezaei@pnu.ac.ir;Tel.:(+989131405381)

Abstract Inthispaper,weintroducethenotionofcommutatoroftwoelementsinaspecificNeutroGroup. ThenwedefinethenotionofaNeutroNilpotentGroupandwestudysomeoftheirproperties.Moreover,we showthattheintersectionoftwoNeutroNilpotentGroupsisaNeutroNilpotentGroup.Also,weshowthatthe quotientofaNeutroNilpotentGroupisaNeutroNilpotentGroup.Specially,usingNeutroHomomorphismwe provetheNeutroNilpotentcyisclosedwithrespecttohomomorphicimage.

Keywords: NeutroGroup;NeutroSubgroup;NeutroNilpotentGroup;NeutroQuotientGroup;NeutroGroupHomomorphism.

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1. Introduction

Oneofthemostimportantconceptsinthestudyofgroupsisthenotionofnilpotency[6]. NilpotentgroupsaroseinGaloistheory,aswellasintheclassificationofgroups.ByGalois theory,certainproblemsinfieldtheoryreducedtogrouptheory.In[10,11],SmarandacheintroducedthenotionsofNeutroDefined,AntiDefinedlaws,NeutroAxiomandAntiAxiom.Then in[9],hestudiedNeutroAlgebrasandAntiAlgebras.Rezaeietal.in[5],provedthatthereare (2n 1)NeutroAlgebrasand(3n 2n)AntiAlgebrasinaclassicalalgebra S with n operations andaxiomsalltogether,where n ≥ 1.Agboolaetal.in[1],studiedNeutroGroups(NG, ∗) wherethelawofcompositionandaxiomsdefinedon NG mayeitherbeonlypartiallydefined (partiallytrue),orpartiallyundefined(partiallyfalse),ortotallyundefined(totallyfalse)with respectto ∗.Moreover,theyconsideredthreeNeutroAxioms(NeutroAssociativity,existence ofNeutroNeutralelementandexistenceofNeutroInverseelement)toshowthedifferencebetweengroupsandNeutroGroups.Also,in[3],AgboolastudiedNeutroRingsbyconsidering threeNeutroAxioms(NeutroAbelianGroup(additive),NeutroSemigroup(multiplicative)and ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups

NeutroDistributivity(multiplicationoveraddition)).ScholarsappliedthenotionofNeutroAxiomsandNeutroLawonRings,Subgrings,Ideals,QuotientRingsandRingHomomorphism topresentsomenewnotionsandseveralresultsareobtained(see[3],[7]).Inthispaper,we consideraclassofNeutroGroupswasintroducedin[1],anddefinethenotionofNeutroNilpotentGroups.Moreover,weinvestigateelementarypropertiesofNeutroNilpotentGroups. Specially,weshowthattheintersectionoftwoNeutroNilpotentGroupsisaNeutroNilpotentGroup.Also,weprovetheNeutroNilpotencyisclosedwithrespecttohomomorphicimage.

2. Preliminaries

Werecallsomebasicdefinitionsandresultswhichareproposedbythepioneersofthis subject.

Definition2.1 ([8]). (i) Aclassicaloperationiswelldefinedforalltheset s elements.

(ii) ANeutroOperationisanoperationpartiallywelldefined,partiallyindeterminate,and partiallyouterdefinedonthegivenset.

(iii) Aclassicallaw/axiomdefinedonanonemptysetistotallytrue(i.e.trueforallset s elements).

(iv) ANeutroLaw/NeutroAxiom(orNeutrosophicLaw/NeutrosophicAxiom)definedona nonemptysetisalaw/axiomthatistrueforsomeset s elements(degreeoftruth (T)),indeterminateforotherset s elements(degreeofindeterminacy(I)),orfalse fortheotherset s elements(degreeoffalsehood(F)),where T,I,F ∈ [0, 1],with (T,I,F ) =(1, 0, 0)thatrepresentstheclassicalaxiom.

(v) ANeutroAlgebraisanalgebrathathasatleastoneNeutroOperationoroneNeutroAxiom(axiomthatistrueforsomeelements,indeterminateforotherelementsand falseforotherelements).

Definition2.2

G andabinaryoperation ∗ onGthecouple(G, ∗) iscalledaclassicalgroupifthefollowingconditionshold:

,where e istheneutral elementof G.

Ifforall x,y ∈ G,(G5) x ∗ y = y ∗ x, then(G, ∗)iscalledanabeliangroup. Notethat x ∗ y willbewrittenas xy forall x,y ∈ G

ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups

Definition2.3 ([6]). Agroup(G, ∗)iscallednilpotentifithasacentralseries,thatis,a normalseries e = G0 ≤ G1 ≤···≤ Gn = G suchthat Gi+1/Gi iscontainedinthecenterof G/Gi forall i. Thelengthofashortestcenteralseriesof G isthenilpotentclassof n Definition2.4 ([6]). Let(G, ∗)beagroupand x1,... , xn beelementsof G.Commutatorof x1 and x2 is[x1,x2]= x 1 1 x 1 2 x1x2. Acommutatorofweight n ≥ 2isdefinedby[x1,...,xn]= [x1,...,xn 1],xn],wherebyconvention[x1]= x1.

ANeutroGroupisanalternativeofagroupthathaseitheroneNeutroOperation(partially well-defined,partiallyindeterminateandpartiallyouter-defined),oratleastoneNeutroAxiom (NeutroAssociativity,NeutroNeutralElementorNeutroInverseElement)withnoAntiOperation(isanoperationouter-definedforalltheset s elements(totallyfalsehood))orAntiAxion (isanaxiomthatisfalseforallset s elements).ItispossibletodefineNeutroGroupinanother waybyconsideringonlyoneNeutroAxiomorbyconsideringtwoNeutroAxiomsoretc.

Definition2.5. Let NG beanonemptysetand ∗ beabinaryoperationon NG.Thecouple (NG, ∗)iscalledaNeutroGroupifthefollowingconditionsaresatisfied:

(NG1) Thereexistssometriplet(x,y,z) ∈ NG suchthat x∗(y ∗z)=(x∗y)∗z and u∗(v ∗w) = (u ∗ v) ∗ w forsome(u,v,w) ∈ NG orthereexistssome(r,s,t) ∈ NG suchthat r ∗ (s ∗ t)=indeterminateor(r ∗ s) ∗ t =indeterminate(NeutroAssociativity).

(NG2) Thereexistsatleastanelement a ∈ NG thathasasingleneutralelementi.e.,wehave e ∈ NG suchthat a ∗ e = e ∗ a = a andfor b ∈ NG theredoesnotexist e ∈ NG suchthat b ∗ e = e ∗ b = b orthereexists e1,e2 ∈ NG suchthat b ∗ e1 = e1 ∗ b = b or b ∗ e2 = e2 ∗ b = b with e1 = e2 orthereexistsatleastanelement c ∈ NG thatthere is d ∈ NG suchthat c ∗ d = d ∗ c =indeterminate(NeutroNeutralElement).

(NG3) Thereexistsanelement a ∈ NG thathasaninverse b ∈ NG w.r.t.aunitelement e ∈ NG i.e., a ∗ b = b ∗ a = e,orthereexistsatleastoneelement b ∈ NG thathas twoormoreinverses c,d ∈ NG w.r.t.someunitelement u ∈ NG i.e., b ∗ c = c ∗ b = u, b ∗ d = d ∗ b = u orthereexistsatleastoneelement r ∈ NG thathasoneelement s ∈ NG suchthat r ∗ s = s ∗ r=indeterminate(NeutroInverseElement).

(NG4) Thereexistssomeduplet(a,b) ∈ NG suchthat a ∗ b = b ∗ a andthereexistssome duplet(c,d) ∈ NG suchthat c ∗ d = d ∗ c,orthereexistssome(r,s) ∈ NG, r ∗ s =indeterminateor s∗r =indeterminate,then(NG, ∗)iscalledaNeutroAbelianGroup (NeutroAbelianGroup).

Example2.6. Let U = {a,b,c,d,e,f } beauniverseofdiscourseand NG = {a,b,c,d} bea subsetof U .Definetheoperation ∗1 on NG intable1.Then ∗1 isaNeutroLowsince c ∗1 d = indeterminate.Also, a ∗1 (b ∗1 c)=(a ∗1 b) ∗1 c and c ∗1 (a ∗1 c)= c ∗1 d =indeterminate

ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups

Table1. ThetableofNeutoGroup(NG, ∗1) ∗1 abcd a bcda b cdac c dab? d ab?a

Thus,(NG, ∗1)isaNeutroGroup. Notethat x ∗ y willbewrittenas xy forall x,y ∈ NG

Theorem2.7 ([1]). Let (NH, ∗) beaNeutroSubgroupoftheNeutroGroup (NG, ∗).Thesets (NG/NH)l = {xNH : x ∈ NG} and (NG/NH)r = {NHx : x ∈ NG} aretwoNeutroGroups withoperations ◦l, ◦r whereforany xNH,yNH ∈ (NG/NH)l, NHx,NHy ∈ (NG/NH)r, x,y ∈ NG wehave xNH ◦l yNH = xyNH,NHx ◦r NHy = NHxy. Definition2.8 ([1]). Let(NG, ∗)and(NK, ◦)betwoNeutroGroups.Themapping ϕ : NG → NK iscalledaNeutroGroupHomomorphismifforeveryduplet(x,y) ∈ G, wehave ϕ(x ∗ y)= ϕ(x) ◦ ϕ(y).

Inaddition,if ϕ isaNeutroBijection,then ϕ iscalledaNeutroGroupIsomorphism.NeutroGroupEpimorphism,NeutroGroupMonomorphism,NeutroGroupEndomorphismaredefinedsimilarly.

Theorem2.9 ([1]). Let (NG, ∗) and (NK, ◦) beNeutroGroupsandlet eNG and eNH be NeutroNeutralElementsin NG and NK respectively.Supposethat ϕ : NG → NK isa NeutroGroupHomomorphism.Then ϕ(eNG)= eNK .

Fromnowon, NG isaNeutroGroupwithtreeNeutroAxioms(NeutroAssociativity,NeutroNeutralElementandNeutroInverseElement).Also,forall x ∈ NG, Nx and Ix represent theNeutroNeutralElementandtheNeutroInverseElementrespectively.

3. SomeResultsOnNeutroNilpotentGroups

Inthissection,weintroducethenotionofcommutatoroftwoelementsinaNeutroGroup andstudyanewconceptas NeutroNilpotentGroups andtheirpropertiesaregiven. Let x,y beelementsofaNeutroGroup NG.Thecommutatorof x,y,denotedby[x,y],is theelement IxIy xy,i.e.,[x,y]= IxIy xy.If Ix or Iy doesnotexist,thenput Ix = x and Iy = y Also,forany x,y1,...,yn ∈ NG,definethecommutator[x,y1,...,yn]by[x,y1,...,yn]= [[x,y1,...,yn 1],yn]. ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups

Table2. ThetableofNeutoNilpotentGroup(NG, ∗2) ∗2 abcd a bcda b cdac c dabd d abca Table3. ThetableofNeutoAbelianGroup(NG, ∗3) ∗3 abce a baba b cfcb c dcec e abce

Definition3.1. ANeutroGroup(NG, ∗)iscalled NeutroNilpotentGroup if Zn(NG)= NG forsome n ∈ N,where Zn(NG)= {x ∈ NG :[x,g1,g2,...,gn]= Nz foratleastone g1,...,gn,z ∈ NG}

Thesmallestsuch n iscalledtheNeutroNilpotencyof NG Notethat,if NG isaNeutroNilpotentGroup,thenforany x ∈ NG thereexistsatleastone g1,...,gn,z ∈ NG suchthat[x,g1,g2,...,gn]= Nz

Example3.2. Let U = {a,b,c,d,e,f } beauniverseofdiscourseand NG = {a,b,c,d} bea subsetof U .Definetheoperation ∗2 on NG intable2.Since[a,b]= d,[d,b]= d,[c,c]= d and[b,b]= a,wehave[c,d,b]=[b,b,b]=[a,b]= d = Na,[d,b,b]=[d,b]= Na and [a,b,b]=[d,b]= Na.Therefore, NG isaNeutroNilpotentGroupofclass2.

Example3.3. Let U = {a,b,c,d,e,f } beauniverseofdiscourseandlet NG = {e,a,b,c} be asubsetof U .Definetheoperation ∗3 on NG intable3.Since[a,b,a]= e = Nc,[b,a,e]= [e,e]= e,[c,e,c]=[c,c]= e,[e,a,a]=[b,a]= e,wehave NG isNeutroAbelianGroupanda NeutroNilpotentGroupofclass2.

InwhatfollowswehaveanonAbelianNeutroNilpotentGroup.

Example3.4. Let U = {a,b,c,d} beauniverseand NG = {a,b,c} beaNeutroGroupbythe Cayleytable4.Then H = {a,b},bytheoperation ∗4,isaNeutroSubgroupof NG (see[1]). Since Na = a, Ia = a, Nb, Ib doesnotexist,wehave[a,a]= aaaa = a = Na and[b,b]= a = Na Therefore, H isaNeutroNilpotentSubgroupthatisnotanAbelianNeutroGroup. ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups

Table4. ThetableofnonAbelianNeutoAbelianGroup(NG, ∗4)

∗4 abc a acb b cac c acd

Table5. ThetableofNeutoSubgroup(H, ∗4) ∗4 ab a ac b ca Table6. ThetableofNeutoSubgroup(NH, ∗5)of(NG, ∗5) ∗5 acd a bda c dbd d aca

Theorem3.5. Let NG and NK betwoNeutroGroups.Then Zn(NG × NK)= Zn(NG) × Zn(NK).Moreover, NG × NK isaNeutroNilpotentGroupofclass n ifandonlyif NG and NK areNeutroNilpotentGroupsofclass n.

Proof. Assume(x,y) ∈ Zn(NG × NK), z ∈ NG and t ∈ NK.Thenforsome (x1,y1),..., (xn,yn) ∈ NG × NK,wehave (Nz ,Nt)=[(x,y), (x1,y1),..., (xn,yn)]=([x,x1,...,xn], [y,y1,...,yn]) ⇔ [x,x1,...,xn]= Nz , [y,y1,...,yn]= Nt ⇔ x ∈ Zn(NG),y ∈ Zn(NK) ⇔ (x,y) ∈ Zn(NG) × Zn(NK) Therefore, Zn(NG × NK)= Zn(NG) × Zn(NK). Moreover, NG × NK isNeutroNilpotentGroupifandonlyif Zn(NG × NK)= NG × NK = Zn(NG) × Zn(NK)ifandonlyif NG and NK areNeutroNilpotentGroups.

InwhatfollowswehaveaNeutroSubgroupthatisnotNeutroNilpotentGroup.

Example3.6. ConsidertheNeutroGroup NG fromExample3.2.Definetheoperation ∗5 on NG intable6.Then NH = {a,c,d} isaNeutroSubgroupof NG (see[1]).Since[a,d]= a, [a,a]and[a,c]doesnotexist,weget[a,g1,...,gn]doesnotexistforany g1,...,gn ∈ NH, andso a ∈ Zn(NH)i.e., NH isnotNeutroNilpotentGroup. ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups

Theorem3.7. Let NH beaNeutroSubgroupoftheNeutroNilpotentGroup NG.ThenNeutroQuotientGroups (NG/NH)l and (NG/NH)r areNeutroNilpotentGroups.

Proof. Assume NH beaNeutroSubgroupof NG and gH ∈ (NG/NH)l.Since NG isa NeutroNilpotentGroup,wehave[g,g1,...,gn]= Nz forsome g1,... , gn,z ∈ NG,and so[gNH,g1NH,...,gnNH]=[g,g1,...,gn]NH = Nz NH.Since(zNH) ◦l (Nz NH)= (z ∗ Nz )NH = zNH =(Nz )NH ◦l zNH,weget(Nz )NH isaNeutroNaturalElementof (NG/NH)l.Therefore,(NG/NH)l isaNeutroNilpotentGroup.Similarly,(NG/NH)r isa NeutroNilpotentGroup.

WerecallthattheintersectionoftwoNeutroGroupsisaNeutroGroup(see[1]).Nowwe havethefollowing:

Theorem3.8. Let NG and NK betwoNeutroNilpotentGroups.Then NG ∩ NK isaNeutroNilpotentGroup.

Proof. Straightforward.

Theorem3.9. Let NH beaNeutroNilpotentSubgroupofaNeutroGroup NG andforall x,t ∈ NG wehave xNH = NH ⇒ x ∈ NH, (Nt)NH = NH.

If (NG/NH)l isaNeutroNilpotentQuotientGroup,then NG isaNeutroNilpotentGroup.

Proof. Assume(NG/NH)l isNeutroNilpotentGroupofclass n and NH isNeutroNilpotentGroupofclass m.Thenforany xNH ∈ (NG/NH)l,thereexist g1NH,...,gnNH ∈ (NG/NH)l suchthat[xNH,g1NH,...,gnNH]=(Nz )NH,where z ∈ NG.Then [x,g1,...,gn]NH =(Nz )NH = NH,andso[x,g1,...,gn] ∈ NH.Since NH isNeutroNilpotentGroup,wegetthereexist k1,...,km ∈ NH suchthat[[x,g1,...,gn],k1,...,km]= Nt, forsome t ∈ NG.Consequently, NG isNeutroNilpotentGroupofclass n + m.

Theorem3.10. Let NH beaNeutroNilpotentSubgroupofaNeutroGroup NG andforall x,t ∈ NG wehave

NHx = NH ⇒ x ∈ NH,NH(Nt)= NH.

If (NG/NH)r isaNeutroNilpotentQuotientGroup,then NG isaNeutroNilpotentGroup.

Proof. SimilartotheproofofTheorem3.9.

Theorem3.11. EveryhomomorphicimageofaNeytroNilpotentGroupisNeutroNilpotentGroup.

ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups

Proof. Assume NH beaNeutroSubgroupofaNeutroNilpotentGroup NG and e1,e2 be NeutroNeutralElementsin NG and NH,respectively.Supposethat ψ : NG → NH isaNeutroGroupEpimorphism.Thenforany h ∈ NH,thereexists x ∈ NG such that h = ψ(x).Since NG isNeutroNilpotentGroup,for x ∈ NG,thereexist g1,..., gn ∈ NG suchthat[x,g1,...,gn]= e1.Take k1 = ψ(g1),...,kn = ψ(gn).Therefore, [h,k1,...,kn]= ψ([x,g1,...,gn])= ψ(e1)= e2,andso NH isaNeutroNilpotenGroup.

4. Conclusion

Inthispaper,wedefinedaclassofNeutroGroups,namedNeutroNilpotentGroups,and theirelementarypropertieswerepresented.TheintersectionoftwoNeutroSubgroupsisnot necessarilyaNeutroSubgroupwhiletheirunionisaNeutroSubgroup.Wehopetostudy NeutroSolvabelGroups,NeutroEngelGroupsinourfutureworks.

Acknowledgments: TheauthorswishtothankEditors-in-Chief,andaregratefultoProf. A.A.A.Agboola,andtherefereesfortheirvaluablesuggestionsanddiscussionsonthiswork.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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Received: June 29, 2020. Accepted: Nov 20, 2020

ElaheMohammadzadeh,AkbarRezaei,OnNeutroNilpotentGroups