A thesis submitted in fulfillment of the requirements for the degree of MASTER OF SCIENCE IN MATHEMATICS

Transcription

1 THE CLASSIFICATION OF SOME FUY SUBGROUPS OF FINITE GROUPS UNDER A NATURAL EQUIVALENCE AND ITS EXTENSION, WITH PARTICULAR EMPHASIS ON THE NUMBER OF EQUIVALENCE CLASSES A thesis subitted i fulfillet of the euieets fo the degee of MASTER OF SCIENCE IN MATHEMATICS I the Faculty of Sciece ad Agicultue at the Uivesity of Fot Hae BY ODILO NDIWENI Octobe 7 Nae of Sueviso: Pof BB Maaba

2 Abstact I this thesis we use the atual euivalece of fuzzy subgous studied by Muali ad Maaba [5] to chaacteize fuzzy subgous of soe fiite gous We focus o the deteiatio of the ube of euivalece classes of fuzzy subgous of soe selected fiite gous usig this euivalece elatio ad its extesio Fistly we give a bief discussio o the theoy of fuzzy sets ad fuzzy subgous We ove a few oeties of fuzzy sets ad fuzzy subgous We the itoduce the selected gous aely the syetic gou S, dihedal gou D, the uateio gou Q 8, cyclic -gou G, G, G ad G s whee, ad ae distict ies ad,, s N We also eset thei subgous stuctues ad costuct lattice diagas of subgous i ode to study thei axial chais We coute the ube of axial chais ad give a bief exlaatio o how the axial chais ae used i the deteiatio of the ube of euivalece classes of fuzzy subgous I deteiig the ube of euivalece classes of fuzzy subgous of a gou, we fist list dow all the axial chais of the gou Secodly we ic ay axial chai ad coute the ube of distict fuzzy subgous eeseted by that axial chai, exessig each fuzzy subgou i the fo of a eychai Theeafte we ic the ext axial chai ad cout the ube of euivalece classes of fuzzy subgous ot couted i the fist chai We oceed iductively util all the axial chais have bee exhausted The total ube of fuzzy subgous obtaied i all the axial chais eesets the ube of euivalece classes of fuzzy subgous fo the etie gou, see sectios,,, 8, 9, 5, ad 7 fo the case of selected fiite gous We study, establish ad ove the foulae fo the ube of axial chais fo the gous G, G ad G s whee, ad ae distict ies ad,, s N To accolish this, we use lattice diagas of subgous of these gous to idetify the axial chais Fo istace, the gou G would euie the use of a - diesioal ectagula diaga see sectio 8 ad 55, while fo the gou G we execute - diesioal lattice diagas of subgous see sectio 5, 5, 5, 55 ad 5 It is though these lattice diagas that we idetify outes though which to cay out the extesios Sice fuzzy subgous eeseted by axial chais ae viewed as eychais, we give a bief discussio o the otio of eychais, is ad thei extesios We eset oositios ad oofs o why this coutig techiue is justifiable We deive ad ove foulae fo the ube of euivalece classes of the gous G, G ad G s whee, ad ae distict ies ad,, s N We give a detailed exlaatio ad illustatios o how this eychai extesio icile wos i Chate Five We coclude by givig secific illustatios o how we coute the ube of euivalece classes of a fuzzy subgou fo the gou G fo the ube of fuzzy subgous of the gou G This illustates a geeal techiue of coutig the ube of fuzzy subgous of s

3 G s fo the ube of fuzzy subgous of s G Ou illustatio also shows two ways of extedig fo a lattice diaga of G to that of G KEY WORDS: Fuzzy Subgous, oal fuzzy subgous, axial chais, euivalet fuzzy subgous, eychais, ode ad i extesio

4 CONTENTS PAGE Abstact Acowledgeet Itoductio 7 Chate : Fuzzy Sets, Fuzzy Subgous, Fuzzy Noal Subgous Itoductio Fuzzy Sets Fuzzy Subgous Fuzzy Noal Subgous Chate : Fuzzy Euivalece Relatios ad Fuzzy Isoohis Itoductio 5 A euivalece elatio 5 Fuzzy elatios 5 Fuzzy euivalece elatio Fuzzy isoohis 8 Chate : O Euivalece of Fuzzy Subgous, Isoohic Classes of Fuzzy Subgous of selected fiite gous Itoductio Euivalet Fuzzy Subgous Classificatio of Fuzzy Subgous of Fiite gous Isoohic Classes of Fuzzy Subgous 7 Chate : O the Maxial chais of the gous G ad G s Itoductio 5 Maxial Chais of G 5

5 5 5 Maxial chais of G 5 Maxial chais of G s 57 Chate 5: O the Nube of euivalece classes of Fuzzy Subgous fo the gous G ad G s 5 Itoductio 7 5 Keychais ad i-extesios 7 5 Justificatio of the Coutig techiue of Fuzzy Subgous 7 5 Classificatio of fuzzy subgous of G 8 5 Classificatio of fuzzy subgous of G 9 55 Coclusio Refeeces

6 Acowledgeets I owe gatitude to Pof BB Maaba ot oly fo atietly etoig e thoughout the couse of y studies but also affodig e fiacial assistace by offeig e a tutoshi ositio i the deatet, fo this I a deely ideted to hi I also tha vey uch Pof S Mabizela ad Pof V Muali fo helig e access the Rhodes Uivesity Libay May thas also go to ebes of staff at the Deatet of Matheatics at the Uivesity of Fot Hae fo thei ivaluable suot ad ecouageet duig y study Fially I tha God fo aig it ossible fo to udetae this eseach, ad fo givig e the deteiatio to colete y studies

7 7 Itoductio Hua beigs baely coehed uatitatively soe decisio-aig ad oble- solvig tass that ae colex, hece the eed fo the executio of owledge that is iecise to each defiite decisios This has led to the advet of fuzzy set theoy thought to eseble hua easoig i its use of aoxiate data ad ucetaity i the geeatio of decisios Although Fuzzy Logic dates bac to Plato, Luaieviz 9s at soe stage efeed to it as May-Valued logic, it was foalized by Pofesso Lotfi adeh i the 9s The te Fuzzy Logic is ebacive as it is used to descibe the lies of fuzzy aithetic, fuzzy atheatical ogaig, fuzzy toology, fuzzy logic, fuzzy gah theoy ad fuzzy data aalysis which ae custoaily called Fuzzy set theoy This theoy of fuzzy subsets as develoed by adeh L has a wide age of alicatios, fo exale it has bee used by Rosefield i 97 to develo the theoy of fuzzy gous Othe otios have bee develoed based o this theoy, these iclude aog othes, the otio of level subgous by PS Das used to chaacteize fuzzy subgous of fiite gous ad the otio of Euivalece of fuzzy subgous itoduced by Maaba ad Muali which will be used i this thesis I this thesis we use this atual euivalece to study the chaacteizatio of soe fiite gous, we coae the ube of euivalece classes ad isoohic classes of these secific gous It was i 97 that Rosefeld [ ] fist ublished his wo o fuzzy gous PS Das[ ], Muhejee ad Bhattachaya [ 7 ] followed a decade late The latte chaacteized fuzzy subgous executig the otios of fuzzy cosets ad fuzzy oal subgous Das[ ] itoduced level subgous ad chaacteized fuzzy subgous of fiite gous by thei level subgous, he oved that they fo a chai He aised the oble of fidig a fuzzy subgou that is eesetative of all the level subgous This oble was asweed by Bhattachaya[ 5 ], he aaged to show that give ay chai of subgous of a fiite gou thee exists a fuzzy subgou of that gou whose level subgous ae ecisely the ebes of that fiite chai A iotat discovey by [ 5 ] was that this fuzzy subgou is ot uiue, i othe wods two distict fuzzy subgous ca have the sae faily of level subgous We use this chaacteizatio i this thesis The sae autho i [ ] oves that two fuzzy

8 8 subgous of fiite gous with idetical level subgous ae eual if ad oly if thei iage sets ae eual Bhattachaya i [ ] also geealized Rosefeld[, 5] Theoe ad Das [, 5] Theoe Fuzzy oality was itoduced by Bhattachaya ad Muhejee i[ 7 ] Seveal studies o the cocet have bee doe by[ ], [ ], [ ],[ 7 ],[ ] ad [ ] just to etio a few Fo istace Agul[ ] studied fuzzy oality, fuzzy level oal subgous ad thei hooohis Maaba ad Muali i [ ] oved that oal fuzzy subgous ad coguece elatios deteie each othe i a gou theoetical sese Shewood[ 8 ] itoduced the cocet of exteal diect oduct of fuzzy subgous Maaba [ ] itoduced the cocet of iteal diect oduct ad oved that both ae isoohic if the fuzzy subgous ae fuzzy oal Rosefeld[ ] oved that a hooohic iage of a fuzzy subgou is a fuzzy subgou ovided the fuzzy subgou has a su-oety, while a hooohic eiage of a fuzzy subgou is always a fuzzy subgou Athoy ad Shewood [ ] late oved that eve without the su-oety the hooohic iage of a fuzzy subgou is a fuzzy subgou Othe studies o hooohic iages ad e-iages of fuzzy subgous wee doe by Sidy ad Mishef, Kua[ 9 ], Abou-aid[ ], Maaba[ ] ad Muali [ ] The otio of a fuzzy elatio was fist defied o a set by adeh[ 9,], futhe studies wee accolished by Rosefeld[ ] ad Kaufa[ ] Foato, Scaati ad Gela[ ] ad adeh[ ] also studied siilaity elatio, which we do ot esue i this thesis Chaaboty ad Das[ 9,] studied fuzzy elatio i coectio with euivalece elatios ad fuzzy fuctios Muali ad Maaba[ 5,,7,8] istead studied fuzzy elatios i coectio with atitios ad deived a suitable atual euivalece elatio o the class of all fuzzy sets of a set This they used to chaacteize ad deteie the ube of distict euivalece classes of fuzzy subgous of -gous Muali ad Maaba i [ ] chaacteize fuzzy subgous of soe fiite gous by use of eychais The sae authos i [ 7 ] itoduced the otio of a ied flag i ode to study the oeatios su, uio ad itesectio i elatio to this atual euivalece

9 9 Thee have bee a ube of studies ivolvig the use of this euivalece elatio, see fo exale Muali ad Maaba[ 8,9] ad Ngcibi[ ] I Chate we defie a fuzzy set i geeal ad chaacteize fuzzy sets usig α cuts We itoduce the otio of a fuzzy subgou ad give a few oeties of fuzzy subgous We give the defiitio of a oduct of fuzzy subgous as give by adeh[ 9 ] ad Maaba[ ] We also study fuzzy oality, its chaacteizatio by level subgous ad fuzzy oits We colude the Chate by ovig that if µ is a fuzzy subgou of a gou the the hooohic iage f µ ad hooohic eiage ae fuzzy subgous of the sae gou I Chate the otio of a fuzzy euivalece elatio is itoduced see Muali[ ], Muali ad Maaba[ 5 ],[ ],[ 7 ], Ngcibi[ ] I [ ] Muali defied ad studied oeties, icludig cuts, of fuzzy euivalece elatios o a set It is the atual euivalece elatio itoduced by Muali ad Maaba fo oe details see[ 5 ],[ ] ad [ 7 ] that we ae goig to extesively use i this thesis We give this defiitio give also by Mual ad Maaba ad show that it is ideed a euivalece elatio We also defie a cotiuous ad biefly discuss the usefuless of t o, chaacteize a t o that is t o A bief discussio o the euivalece of fuzzy subgous ad soe coseueces is give i this chate Secific exales ae give o euivalet ad o-euivalet fuzzy subgous We chaacteize euivalece betwee fuzzy subgous usig level subgouswe colude the chate with a bief discussio o hooohic iages ad e-iages Faleigh[ ] chaacteizes fiite Abelia gous i the cis case Muali ad Maaba i[ 5 ], [ ] ad [ 7 ] studied the classificatio of fuzzy subgous of fiite Abelia gous usig diffeet aoaches that iclude the ube of o-euivalet fuzzy subgous fo the gou ad G whee ad ae distict ies, i[ 5 ] I [ ] they ivestigated the ube of fuzzy subgous of G fo distict ies i fo i,,,, ad also distict fuzzy subgous of G, whee ad ae distict ies, N ad

10 ,,,,5 wee also studied Ngcibi[ ] also used the otio of euivalece of fuzzy subgous studied by Muali ad Maaba to chaacteize fuzzy subgous of -gous fo secified ies The autho[ ] also did a classificatio of fuzzy subgous of Abelia gous of the fo G ad of the fo G fo the cases ad I Chate we itoduce soe secific gous, aely the syetic gou S, dihedal gou D,the uateio gou Q 8, cyclic -gou G ad the gou G We eset subgous, lattice stuctue of subgous ad axial chais It is i this chate that we give the defiitio of fuzzy isoohis give by Muali ad Maaba[ 5 ], we deteie the ube of distict fuzzy subgous ad isoohic classes of fuzzy subgous fo these gous Coaisos ae ade o the ube of distict fuzzy subgous ad the ube of isoohic classes Foulae fo the ube of distict fuzzy subgous fo selected gous give by Muali ad Maaba i[ 5 ],[ ] ad [ 7 ] ad Ngcibi[ ] ae also veified o these gous we ae studyig I Chate we defie a axial subgou of a gou ad deostate with a few lattice diagas the deteiatio of the ube of axial chais We establish ad give oofs, i the fo of leas ad oositios, of foulae fo the ube of axial chais fo thegous G, G ad G s whee,, ae distict ies ad,, s N Chate 5 is a extesio of chate Havig obtaied the foulae fo the ube of axial chais fo the gous, we go futhe o ad itoduce the otios of eychais, is, ied-flag fo oe see Muali ad Maaba[ 5 ],[ ] ad [ 7 ] ad i extesio which we exloit i the coutatio of the ube of euivalece classes of fuzzy subgous fo these gous We give a detailed exlaatio of the ethod of coutig the ube of fuzzy subgous usig axial chais This we accolish by statig the coutig techiue i tes of oositios Secific exales ae give to illustate how the coutig techiue is alied

11 I 5 we iclude soe wo by Ngcibi[ ] o the foulae fo the distict ube of fuzzy subgous fo the gou G whee, ae distict ies ad,, We also give a oof of Ngcibi s Theoe 5 i [ ] which the autho did ot ove This we do as aothe illustatio fo the justificatio of ou coutig techiue We list a few cobiatoial aalysis defiitios that ae used i this oof fo oe see Rioda [ ] We establish ad give oof, with a aid of - diesioal lattice diagas, of foulae fo the ube of distict fuzzy subgous of the gou G whee,, ae distict ies ad N,, s ad s We coclude by showig how i geeal the ube of distict fuzzy subgous of G ca be obtaied if the ube of distict fuzzy subgous of s H s o s with a secific case o s is ow, illustatig

12 CHAPTER ONE Fuzzy Sets, Fuzzy Subgous, Fuzzy Noal Subgous Itoductio I ode to study fuzzy subgous, the theoy of fuzzy sets is exteded ad alied to the gou stuctual settigs I this toic we give a eliiay discussio o the geeal oeties of fuzzy sets ad chaacteize fuzzy sets usig alha- cuts The otio of fuzzy subgous as defied by Rosefeld[ ] is give ad a few oeties of fuzzy subgous oved adeh[ 9 ] ad Maaba[ ] defied the oduct of two fuzzy subgous, this defiitio is give i this chate The otio of level subgous has bee used by seveal eseaches i the classificatio of fuzzy subgous, icludig aog othes, Das[ ], Bhattachaya[ ], ad Maaba[ ] Fuzzy oality is studied ad chaacteized usig level subgous ad fuzzy oits We coclude by ovig that if µ is a fuzzy subgou of a gou G the the hooohic iage f µ ad hooohic e- iage ae fuzzy subgous of the sae gou Siila esults wee obtaied by Rosefeld[ ], Kua[ 9 ] ad Maaba[ ] Fuzzy sets A fuzzy set is a set deived by geealizig the cocet of cis set Ulie i cis set theoy whee thee is total ebeshi, say x belogs to a set U witte as fuzzy sets allow eleets to atially belog to a set A fuzzy subset of a set U is a fuctio [ ] µ : U, x U, If the iage set is {,} the we have a cis set We soeties eeset the fuzzy set : A [,] µ by µ A whee µ A x t fo x A, t We the say t is the degee to which x belogs to the fuzzy subset µ We obseve that whe t, we ea absolute o-ebeshi, ad whe t, absolute ebeshi If µ x < µ y the we say y belogs to µ oe tha x belogs to µ A

13 Oeatios o Fuzzy sets ***Uio of two fuzzy sets µ A ad µ B called the Maxiu Citeio, is defied as µ A B ax µ A, µ B µ A µ B ***Itesectio of two fuzzy sets µ A ad µ B called the Miiu Citeio, is defied as µ A B i µ A, µ B µ A µ B ***Coleet of µ A is defied as C µ A x µ x A ***Iclusio Fix a set U Suose µ ad ν ae two fuzzy sets, µ :U I, ν :U I, the by µ ν we ea µ x ν x x U ***Euality µ ν µ x ν x, x U ***Null set Is descibed by the ebeshi fuctio µ x, x U φ ***Whole set Is the fuzzy set µ x, x U U { µ j x : j J} su µ x ad { µ j x : j J} if µ x j J j j J j

14 Fuzzy Poits Coside a o-ety uivesal set U The set of all fuzzy subsets of U is deoted by I U Defiitio [ ] A fuzzy subset µ : X I is called a fuzzy oit if µ x, x X excet fo oe ad oly oe eleet of X Coseueces of defiitio Fistly µ x fo oe ad oly oe eleet of X Coside a X : µ a The µ a λ, < λ by the defiitio of µ x Case I: If λ the µ x whe x a ad whe x a cis sigleto { a }, the fuzzy set is the Case II: If < λ < the µ x λ whe x a ad othewise b Thus µ is a fuzzy oit ad we deote it by So λ a is such that λ a λ a x λ if x a ad if x a, this ilies that λ a a λ c Fo c suose < λ < λ < λ the a λ λ a a λ Poositio 5[ ] Let X λ λ µ I The µ { a : a µ } O α cuts µ ad α Coside a fuzzy set : X I [,] Defiitio 7 [ ] The wea α cut of µ deoted by µ α is defied as { x X : µ x α} µ α

15 5 Defiitio 8[ ] The stog α cut of µ deoted by { x X : µ x α} µ α > α µ is defied as ***Coseueces of defiitios 7 ad 8 a α µ α φ b α µ α X Defiitio 9[ ] The Suot of µ is defied as follows { x X : > } Su µ µ x Chaacteizatio of fuzzy sets usig α cuts A fuzzy set ca be chaacteized usig α cuts as the followig oositio shows Poositio Give ay fuzzy set µ the µ su αχ < α < αχ µ α µ α dx o µ α αχ µ α µ,, α α αχ Poof Let µ x α, the x µ α α χ µ x α µ x α Now if β > µx, the x µ β βχ x, thus µ x α χ µ x su αχ µ x su αχ x µ β Also give ay fuzzy set µ, µ x αχ µ x dx Poof Let µ x α, the µ x αχ µ x α α α µ α α α αχ µ α x dx sice x µ α α

16 Theefoe x µ αχ x dx µ α Chais of α cuts Suose < α < β < the α β µ µ ad also α µ β µ Coseuetly give a chai of ubes λ λ λ λ, we have µ λ µ λ µ λ µ λ Iages ad e-iages of fuzzy sets [ 7 ] Coside X ad Y to be two uivesal o-ety sets ad f : X Y be a fuctio fo X to Y ad let µ : X I be a fuzzy subset of X By f µ we ea a fuzzy set of Y defied by f su, if { µ x : x f y } µ y x f Thus the degee to which y belogs to f µ is at least as uch as the degee to which x belogs to µ, x fo which f x y y Defiitio: [ 7 ] Let f : X Y be a fuctio If ν is a fuzzy subset of Y the the e-iage f ν is a fuzzy subset of X defied by f ν g ν f g, g X Fuzzy Subgous[ ] A fuzzy subset µ : G I of a gou G is a fuzzy subgou of G if i µ xy i{ µ x, µ y }, x, y G ii µ x µ x, x G Fo the idetity eleet e G, µ x µ e, x G Euivaletly we have

17 7 Poositio: A fuzzy subset µ of G is a fuzzy subgou of G iff a µοµ µ ad b µ µ whee µ is defied as : µ G I, g G, µ g µ g Befoe we give a oof of the above oositio we fist give two iotat defiitios Defiitio: [ ] We defie µοµ g su µ g µ g Defiitio: [ ] g g g If µ is a fuzzy subgou o a gou G ad θ is a a fo G oto itself, we defie θ a a : G [,] whee µ by θ θ µ g µ g, g G θ g is the iage of g udeθ Poof of a Let g, g G be abitay, ow sice µ is a fuzzy subgou ofg, µ g g µ g µ g, set g g g Taig the sueu ove both sides we obtai µ g su µ g µ g su g g g g g g µ g µ g µ µ µ g o g gg Theefoe µ o µ µ b µ is a fuzzy subgou g g µ µ, g G But by defiitio g µ µ g, g G Theefoe µ µ if µ o µ µ ad µ µ, we eed to show that µ is a fuzzy subgou Now µo µ xy µ xy, x, y G ad µοµ xy su{ µ a µ b } xy ab

18 8 xy i{ µ x, µ y } µ { µ x, µ y } i x, y G Sice µ g µ g g G ad µ g µ g g G, the it follows that µ g µ g g G Theefoe µ is a fuzzy subgou of G Defiitio [ ] Let G be a gou ad µ be a fuzzy subgou of G The subgous µ, [,] t t ad t e ae called level subgous of G Defiitio [ ] Let µ ad ν be fuzzy subsets ofg The oduct : G [,] µν x su µ x ν x, x, x x G Poositio: 5 x x x, µν is defied by If µ is a fuzzy subgou of a gou, the xy i µ x, µ y x, y G, µ x µ y Poof see A Mustafa[ ] µ fo each Poeties of fuzzy subgous Utilizig the defiitios give above we coe u with the followig oeties of fuzzy subgous Poositio: If µ is a fuzzy subset of a gou G, the µ is a fuzzy subgou if ad oly if each µ is a subgou of G, t t Poof µ is a fuzzy subgou We eed to show that µ t is a subgou of G Let x, y µ the µ x t ad y t Let t µ xy i x, y t xy µ µ µ µ t x µ t the µ x t µ x µ x t, thus x µ t Theefoe µ t is a subgou ofg

19 t ofg µ is a subgou of G t [,] 9 We eed to show that µ is a fuzzy subgou Let x, y G Fo x µ t ad y µ t we have µ x t ad µ y t But sice µ is a subgou ofg the xy µ t µ xy t t Theefoe xy i µ x, µ y Case If µ x µ t ad y µ s s < t the µ t µ s, so x µ s Thus x, y µ s ad sice µ s is a subgou of G this ilies that xy µ s µ xy i µ x, µ y Siilaly if t < s Let x G Fo x µ t we have x µ t µ x t µ x µ x, x G Thus x µ x µ x µ Hece µ x µ x Theefoe µ is a subgou ofg This coletes the oof Poositio: Let µ be a fuzzy subset of G The µ is a fuzzy subgou ofg a, b, a b µ Poof λ β µ λ β assue µ is a fuzzy subgou Let a, b µ The µ a λ ad µ b β λ β Now µ ab µ a µ b µ a µ b λ β ab λ β a b µ λ β µ let x, y G We eed to show that µ xy µ x µ y Let µ x λ ad µ y β If λ, β the µ xy µ x µ y Now we assue λ, β So x, y xy µ, µ xy λ β µ x µ y λ β µ λ β To show that µ x µ x we oceed as follows: case µ x λ Let µ x λ, the x x x µ, thus xx e µ Now λ µ λ λ λ λ µ λ e, x x µ µ x µ x λ λ is a fuzzy subgou By syety µ x µ x Theefoe µ

20 Theoe: Let f : G G be a hooohis ofg itog If µ is a fuzzy subgou of G, the f µ is a fuzzy subgou ofg Poof We eed to show the two coditios of sectio Sice f is ito theefoe f su µ a a f y y µ if if y y f G f G Suose y f G, the y f G Thus f µ y su µ a su µ a f µ y, y G y f a y f a So f µ y f µ y f µ y f µ y f µ y Suose y f G, the y f G f µ y f µ y Let y y y, we ai to show that f µ y f µ y f µ y Coside f µ y su µ a, f µ y su µ a y f a y f a ad f µ y su µ a Taigξ >, the y f a f µ y f µ y ξ < µ a µ a fo soe a, a, a : y f a, y f a, a a a ad y y y Now y y y f a f a f a a f ad µ a µ a µ a a µ a This ilies that f µ y f µ y ξ < µ a su µ a f µ y y f a a f µ y f µ y f µ y Sice ξ is abitay Thus f µ is a fuzzy subgou of G Poositio: Let f : G G be a hooohis ad µ a fuzzy subgou of a gou G The f µ is a fuzzy subgou of G Poof f µ a µ f a

21 µ f a µ f a f µ a Fially f ab µ f ab µ f a f b µ f a µ f b f µ a f µ b Theefoe f µ ab f µ a f µ b Fuzzy Noal subgous Defiitio: [ ] If µ is a fuzzy subgou of a gou G, the µ is called a fuzzy oal subgou if µ xy µ yx, x, y G Euivaletly µ is fuzzy oal if ad oly if µ xyx Poof µ y, x, y G Suose µ is fuzzy oal, the µ xy µ yx, x, y G µ xyx µ x yx µ yx x µ y, x, y G Suose µ xyx µ y, x, y G The µ xy µ xyxx µ yx Poositio: If µ, ν ae fuzzy subgous of a gou G ad µ is fuzzy oal, the µν is a fuzzy subgou of G Poof We eed to show the two coditios of defiitio To show that µν xy µν x µν y we let µν x su µ x ν x x xx

22 ad µν y su µ y ν y y y y Let ξ >, x, x, y, y : x xx, y y y ad ξ ξ µν x < µ x ν x ad µν y < µ y ν y The ξ ξ ξ µν x µν y µν x µν y < µ x ν x µ y ν y y µ x yx ν x y µ x µ x y x ν x ν x µν xy by oality of µ ad sice xy xx yx x y Theefoe µν x µν y µν xy sice ξ is abitay Coditio b: Let x G the µν x su µ x ν x x xx su x ν x x x x µ sice µ ad ν ae fuzzy subgous su x x x ν x x x x µ by x oality of µ µν x x µν sice x x x x x x x By syety we also have µν x µν x Theefoe euality holds Poositio: If µ ad ν ae both fuzzy oal subgous of G, the µν is a fuzzy oal subgou of G Poof We eed to show that µν xyx µν y, x, y G µ a µ b µν xyx su xyx ab x ax x bx su µ ν by oality of µ adν µν y sice y x abx x axx bx Thus µν y µν x yx, x, y G µν y µν x xyx x

23 µν xyx Theefoe µν y µν xyx µν is a fuzzy oal subgou of G Poositio: If µ ad ν ae fuzzy subgous of G ad µ is fuzzy oal, the µν νµ Poof νµ x su ν x µ x x x x µ x x x ν x su x x x sice µ is fuzzy oal µν x x µν x sice x xx x x Siilaly µν x νµ x Poositio: 5 Let µ be a fuzzy subgou ofg µ is fuzzy oal if ad oly if each µ t is a oal fuzzy subgou of G, [,] Poof t We eed to show that xµ x t µ, x G t Let h µ t the µ h t µ h µ xhx t t xhx µ, x G, h µ t µ t x µ x Theefoe t µ x x t t µ Let y x µ t x Now y xhx fo soe h µ t The µ y µ xhx µ h t, sice µ is oal This ilies that y t y µ t Theefoe µ x t t µ x µ Thus x µ µ x t t Let x, y G, also set µ x t The x y y µ t µ t sice t µ oal Theefoe y xy µ t µ y xy t µ x, x, y G This ilies that µ yxy µ x The x µ y yxy y µ yxy µ

24 Theefoe µ x µ yxy, x, y G Thus µ is a fuzzy oal subgou of G

25 5 Chate Two FUY EQUIVALENCE RELATION AND FUY ISOMORPHISM Itoductio Relatig objects that ae eceived eual euies the otio of euivalece elatios Studies o the ilicatios of this euivalece elatio o fuzzy subsets of a set wee accolished by a ube of authos, fo exale i [ ] Muali defied ad studied oeties, icludig cuts, of fuzzy euivalece elatios o a set I this chate we fist give a defiitio of a euivalece elatio i geeal ad secodly that of a fuzzy euivalece elatio fo oe see Muali[ ], Muali ad Maaba [ 5 ],[ ] ad [ 7 ], Ngcibi[ ] We study the atual euivalece elatio itoduced by Muali ad Maaba fo oe details see[ 5 ],[ ] ad [ 7 ] ad show that it is ideed a euivalece elatio We study the euivalece of fuzzy subsets of a set as a foudatio to the study of euivalece of fuzzy subgous of a gou G This we accolish by assigig euivalece classes to the fuzzy subgous of that gou The defiitio of a euivalece class of a eleet of a set is give i Soe coseueces of euivalece of fuzzy subgous ae give We also defie a t o, chaacteize a t o that is cotiuous ad biefly discuss the usefuless of t os A Euivalece Relatio Defiitio: A elatio R, o X is a subset D of X X ad we wite xr y x, y D Now R is a euivalece elatio o X if x, y, z X : a xr x, x X Reflexive law b xr y yrx Syetic law c xr y ad yr z xrz Tasitive law Fuzzy Relatios Defiitio: A fuzzy elatio µ betwee eleets of two sets X ad Y is a fuzzy subset of X Y give by µ : X Y I, x, y µ x, y Note: µ x, y is thought as the degee to which x is elated to y The µ defied above is a biay elatio ad is said to be:

26 a Reflexive if µ x, x, x X b Syetic if µ x, y µ y, x, x, y X c Tasitive if µ o µ µ whee µo µ is defied by µ x, z µ z, µ o µ x, y su y z X Ay fuzzy elatio that satisfies a, b ad c is called a fuzzy euivalece elatio o X Fuzzy Euivalece elatio We defie a euivalece elatio o Defiitio: [ 5 ] X I as follows: Let µ ad ν be two fuzzy subgous µ is fuzzy euivalet to ν deoted by µ ν if ad oly if µ x > µ y ν x > ν y ad µ x ν x Clai : Defiitio is a euivalece elatio We have to chec Reflexive law: Clea fo defiitio Syetic law : Need to show that µ ν ν µ Now µ ν µ x > µ y ν x > ν y ad µ x ν x a Itechagig the oles of µ ad ν i a we obtai: µ ν ν β µ β G Tasitive law: Need to show that fo µ, ν, β, I, µ ν ad Now usig a ad the fact that ν β ν x > ν y β x > β y ad ν x β x we obtai µ x > µ y β x > β y ad µ x β x µ β theefoe defies a euivalece elatio o G Defiitio: Let A be a set ad R a euivalet elatio o A, the the euivalece class of is a set{ x A : arx} a A

27 7 Poositio: Let G be a fiite gou ad µ be a fuzzy subgou of G If t i, t j ae eleets of the iage set of µ such that Poof[ ] µ t µ t, the i t j i j t Poositio: ν µ I µ Iν Poof[ ] Defiitio: 5 Let : [,] [,] T be a biay oeatio, the T is called a tiagula o t o if at is associative bt is coutative ct is o-deceasig fo both vaiables d T x, x, x [,] Coseueces of defiitio 5 ***A t o T is called cout if it eseves the least ue boud ***A t o T is called Achiedea if T x, x < x fo ay [,] x 7 Chaacteizatio of a euivalece by a t o T that is cotiuous A euivalece ca be defied as follows: x y, y x x y T T T This is so because the ilicatio is defied by: { z T x, z y} x T y ax Siilaly { z T y, z x} y T x ax T

28 8 8 Usefuless of t os Although the i, uio, oduct ad bouded su oeatos belog to a class of t os, thee ae uiue defiitios fo the itesectio ad ad uio o i dual logic, taditioal set theoy ad fuzzy set theoy This is so because ost oeatos oly behave exactly the sae if the degees of ebeshi ae esticted to the values ad This shows that thee ae othe ways of aggegatig fuzzy sets besides the i ad uio A t o T as give i Defiitio 5 defies a itesectio ad uio of two fuzzy sets µ A ad µ B as follows : µ µ µ, x G i Itesectio T[ x, x ] x A B A B µ µ µ, x G ii Uio T[ x, x ] x A B A B So usig this defiitio we ote that b ad c esue that a decease of the degee of ebeshi to set A o set B will ot ivolve a icease to the degee of ebeshi to the itesectio Syety is also exessed by b, ad a guaatees that the itesectio of ay ube of fuzzy sets ca be efoed i ay ode Aat fo the aleady etioed use, a isoohis t o ca be used to defie a otio of Fuzzy Isoohis Reseaches, aogst the Maaba [ ] ad Muali ad Maaba[ 5 ], studied the ube of distict fuzzy subgous of a gou usig a euivalece elatio ad coaed with the otio of isoohis They oticed that the otio of fuzzy euivalece is fie tha the otio of fuzzy isoohis We theefoe defie fuzzy isoohis as a geealizatio of the euivalece elatio eseted i sectio This will eable us to establish a techiue to calculate the ube of isoohic classes of fuzzy subgous of fiite gous we ae to study i chate thee We stat with defiig a hooohis fo the sae of coleteess Defiitio: Let, ' G ad G, o be gous A aig ' f : G G such that f a b f a o f b, a, b G is called a hooohis

29 9 Defiitio: A hooohis that is also a coesodece is called a isoohis Such a aig is said to eseve the gou oeatio ' ' We will deote two gous G ad G that ae isoohic byg G Theoe: Isoohis is a euivalece elatio o the class of all gous Poof[ ] Defiitio: Let µ ad ν be two fuzzy subgous of gous G ad G ' esectively The we say µ is fuzzy isoohic toν, deoted µ ν a isoohis ' f : G G such that µ x > µ y ν f x > ν f y ad µ x ν f x 5 Hooohis ad Euivalece Euivalece classes of hooohic iages ad e-iages of fuzzy subgous wee ivestigated by Muali ad Maaba i[ 7 ], they discoveed that subgou oety is tasfeed to iages ad e-iages by a hooohis betwee gous They also oted that ieuivalet fuzzy subgous ay have euivalet iages ude a hooohis We ecall that if f : G G' is a hooohis, by f µ we ea the iage of a fuzzy subset µ of G ad is a fuzzy subset of { g : g G, f g '} G ' defied by f µ g' su µ g if f g' Ο ad f µ g' if f g' Ο fo g' G' Siilaly if ν is a fuzzy subset of G ', the e-iage of ν, f ν is a fuzzy subset of G ad is defied by f ν g ν f g I oositios ad 7 we suose that f : G H is a hooohis fo a gou G to H Although a oof of Poositio is give by Muali ad Maaba i [ 7 ] we give a diffeet oof usig the defiitio f µ x su µ a x f a

30 Poositio: [ 7 ] If µ ν the f µ f ν Poof Let f µ f a > f µ f b We eed to show that f ν f a > f ν f b Now sice f is a isoohis, the f x f a x a ad f x f b x b So f µ f a > f µ f b su µ x > su µ x theefoe µ a > µ b f x f a f x f b But µ ν ν a > ν b Theefoe suν x > suν x that is f ν f a > f ν f b ad covesely f a f x f b f x If f µ f x the su µ a µ x f a f x this ilies that ν x sice µ ν This ilies that suν a f ν f x Thus f µ f ν ad covesely Poositio: 7[ 7 ] f a f x If µ ν i H the f µ f ν i G Poof Staightfowad

31 Chate Thee ON EQUIVALENCE OF FUY SUBGROUPS AND ISOMORPHIC CLASSES OF FUY SUBGROUPS OF SELECTED FINITE GROUPS Itoductio Chaacteizatio of fiite gous has bee studied by a ube of eseaches, fo ad Bauslag ad Chadle[ ] Muali ad Maaba[ ] exale Faleigh [ ] 5,[ ] ad [ 7] looed ito euivalece of fuzzy subgous i ode to chaacteize fuzzy subgous of fiite abelia gous Ngcibi[ ] also eloyed the euivalece elatio used by Muali ad Maaba to deteie the ube of distict fuzzy subgous of soe secific -gous I this chate we use this euivalece to study the chaacteizatio of the followig gous: the syetic gou S, dihedal gou D, the uateio gou Q 8, cyclic -gou G ad the gou G We begi by esetig thei subgous, lattices of subgous ad axial chais We also use the defiitio of isoohis give i chate two to deteie the ube of euivalece ad isoohic classes of fuzzy subgous of these gous We the coae the ube of euivalece ad isoohic classes fo the gous Euivalet Fuzzy Subgous Defiitio: Two fuzzy subgous µ ad ν ae said to distict if ad oly if[ µ ] [ ν ], whee [ µ ] ad [ ν ] ae euivalece classes cotaiig µ ad ν esectively Exales of euivalet ad o-euivalet fuzzy subgous Exale: S e, a, a, b, ab a b whee a Let { }, if x e fuzzy sets x if x a, µ a ad ν x 7 if othewise 7 e b ad e is the idetity eleet Defie if if if x e x b othewise Hee su µ suν S ad µ a > µ b but ν a ν b theefoe µ ν Exale: S e, a, a, b, ab a b whee a Let { }, fuzzy sets µ x if ad ν x if if if if if x e x ab othewise x e x ab othewise e b ad e is the idetity eleet Defie

32 Clealy µ ab > µ a iff ν ab > ν a but su µ suν theefoe µ is ot euivalet toν Classificatio of Fuzzy Subgous of Fiite Gous The exales give above deostate the iotace of all the coditios i defiitio I ode to eueate the ube of distict fuzzy subgous ad isoohic classes of secific gous i the sectios to follow, we begi by exlaiig how i geeal, distict fuzzy subgous ca be idetified fo a fixed axial chai of subgous The chai is said to be axial if it caot be efied The defiitios of a eychai, i ad ied-flag ae give i sectio 5 Now give ay axial chai of subgous { } G G G G a, we say that the axial chai has legth, which is the ube of cooets i the axial chai A fuzzy subgou µ ca be eeseted by the followig odeed sybols λ λ λ λ whee the λ ' i s ae eal ubes i [, ] that ae i descedig ode The λ s ae called is We obseve that thee ae is fo this axial chai If we idetify each λ λ µ x λ if if if if x G x G x G i ' x \ { } \ G \ G G i with λ i, we have the fuzzy subgou λ λ λ λ is called a eychai of µ We soeties wite µ λ λ λ λ, thus we idetify µ with its eychai whe the udelyig axial chai of subgous is ow Each Gi is a cooet of the axial chai Exale: a The axial chai { } has two cooets levels We theefoe have the followig distict fuzzy subgous fo this chai:, λ ad b The axial chai { e} B S has thee cooets levels Coesodig to this axial chai thee ae seve distict fuzzy subgous eeseted by the eychais, λ,, λλ λβ, λ, Fuzzy Subgous of the syetic gou S The gou of syeties of thee objects has ode ad is defied as S { e, a, a, b, ab a b} whee a e b,

33 Its subgous ae { e, a a } It has fou axial chais viz B,, B { e, b}, B { e, ab} { e} B S,{ e} B S, { e} B S, B { e, a b}, { } e ad S ad{ e} B S a Fo euatio a each chai is of legth thee, which eas that we ca eeset each fuzzy subgou usig a eychai ** with thee is ***, fo exale µ λβ whee > λ > β o the fist chai if x e µ b β if x S \ B Thus x λ if x B \ { e} if x e ν fo > λ > β the µ ν, thus µ λβ is β if x S \ B If x λ if x B \ { e} actually a class of fuzzy subgous The defiitios of a eychai ** ad i *** ae give i sectio 5 ad 5 esectively Now i coutig the ube of distict euivalece classes of fuzzy subgous fo the etie gou, we coside all the axial chais as follows: Let: µ λβ β x e o the fist chai, that is µ x λ if x B \ { e} ν λβ β if if x S \ B o the secod chai, that is ν x λ if x B \ { e} ξ λβ β if if x S x e x e \ B o the thid chai, that is ξ x λ if x B \ { e} if if x S \ B

34 o the fouth chai, that is τ x λ if x B \ { e} τ λβ β if if x e x S Fo the above discussio we ae able to idetify that µ,ν, ξ ad τ ae distict fuzzy subgous whe cosideig these fou distict chais If the ube of distict euivalece classes of fuzzy subgous is couted fo each axial chai, the the total ube of euivalece classes of fuzzy subgous fo the gou ca be calculated The followig sectio deostates how this fact is used to calculate the ube of euivalece classes of fuzzy subgous of S \ B Techiue fo calculatig the ube of euivalece classes of fuzzy subgous of S : Coside the chai { e} B S i a The ube of distict classes of fuzzy subgous was foud to be eual to seve viz λ λλ λβ λ Each oe of the eychais above is used fo each axial chai i the eueatio of the total ube of fuzzy subgous of the whole gou These esults ae tabulated i the table below Distict Keychais λ λλ λβ λ Total # of distict euivalece classes of fuzzy subgous # of ways each couts if all chais cosideed 9 Thus the ube of distict euivalece classes of fuzzy subgous fo the gou G S is 9

35 5 Now looig at the table above, the class of fuzzy subgou eeseted by the eychai has a cout oe because if we coside each chai, this eychai eesets the sae fuzzy subgou µ x, x S i all the chais of subgous The fuzzy subgou λ couts fou ties because fo the sae λ i all the fou chais x B \ e o x B \ e o x B \ e o x B \ e which ae diffeet sets What this eas is that the sae eychai λ eesets a diffeet class of euivalet fuzzy subgous o diffeet axial chais of subgous Fo the costuctio of fuzzy subgous i sectio with λβ elaced with λ we have: µ a µ a > µ b µ ab µ a ν b > ν a ν a ν ab ν a ξ ab > ξ a ξ a ξ b ξ a b b b τ a b > τ a τ a τ b τ ab Fo the aguet above it is clea that µ,ν,ξ adτ ae distict euivalece classes of fuzzy subgous ude the euivalece we ae executig, hece the cout of fou Siilaly the eychais, λβ ad λ will give a cout of fou The Dihedal gou D The gou of syeties of a suae o the octic, has ode eight To idetify the subgous of this gou we coside the ube of eutatios coesodig to the ways that two coies of a suae with vetices,, ad ca be laced, oe coveig the othe If we basically use ρ fo otatios, µ fo io iages i eedicula bisectos of sides, ad δ i fo diagoal flis we obtai the followig eutatios ρ ρ ρ ρ i i µ µ δ δ Alteatively it ca be thought of as a gou geeated by two eleets s ad such that, s ad s s Thus D {,,,, s, s, s s},

36 Subgous of D The te subgous f D ae listed below: { ρ }, { ρ, δ }, { ρ, δ }, { ρ, ρ }, { ρ, µ }, { ρ, µ }, { ρ, ρ, ρ, ρ }, { ρ, ρ, µ, µ }, { ρ, ρ, δ δ }, ad D I view of the discussio give o subgous of the octic we ae able to costuct axial chais fo this gou i sectio 5 5 Maxial Chais fo D Thee ae seve axial chais fo this gou { ρ } { ρ, δ } { ρ, ρ, δ, δ } D { ρ } { ρ, δ} { ρ, ρ, δ, δ } D { ρ } { ρ, ρ } { ρ, ρ, δ, δ } D { ρ } { ρ, ρ } { ρ, ρ, ρ, ρ} D { ρ } { ρ, ρ } { ρ, ρ, µ, µ } D { ρ } { ρ, µ } { ρ, ρ, µ, µ } D { o} { ρ, µ } { ρ, ρ, µ, µ } D ρ 5a Each chai i 5a is of legth fou A eychai of D is of the fo λβα whee λ β α The ube of euivalece classes of fuzzy subgous fo D I all the chais the distict fuzzy subgou couts oce, that is it eesets oly oe fuzzy subgou µ x, x D The followig table below lists a eychai ad the ube of distict fuzzy subgous it eesets Distict Keychais Nube of couts i all chais λ λλ 5 λβ 7 λ 7

37 7 5 λλλ λλβ λλ λββ 5 λβα 7 λβ 7 λ 5 Total Nube We obtai the above ube of euivalece classes of fuzzy subgous fo each eychai as follows : Usig the axial chais i 5a coside the eychai λβ µ λβ o the fist chai gives µ µ µ µ ρ µ ρ µ δ µ ρ µ δ µ > > ν λβ o the secod chai gives µ ν µ ν ρ ν ρ ν δ ν ρ ν δ ν > > ξ λβ o the thid chai gives µ ξ µ ξ ρ ξ ρ ξ δ ξ δ ξ ρ ξ > > ψ λβ o the fouth chai gives δ ψ δ ψ µ ψ µ ψ ρ ψ ρ ψ ρ ψ > > ϖ λβ o the fith chai gives δ ϖ δ ϖ ρ ϖ ρ ϖ µ ξ µ ϖ ρ ϖ > > τ λβ o the sixth chai gives δ τ δ τ ρ τ ρ τ ρ τ µ τ µ τ > > ς λβ o the seveth chai gives δ ς δ ς ρ ς ρ ς µ ς ρ ς µ ς > >

38 8 Fo the ecedig discussio it is clea that µ, ν, ξ, ψ, ϖ, τ ad ς eeset diffeet euivalece classes of fuzzy subgous whe cosideig all the seve axial chais hece the cout of seve Now i the above costuctio if we elace the eychai µ λββ o the fist chai gives µ δ > µ ρ µ δ µ ρ µ ρ µ µ µ µ ν λββ o the secod chai gives ν δ > ν ρ ν δ ν ρ ν ρ ν µ ν µ ξ λββ o the thid chai gives ξ ρ > ξ δ ξ δ ξ ρ ξ ρ ξ µ ξ µ ψ λββ o the fouth chai gives ψ ρ > ψ ρ ψ ρ ψ µ ψ µ ψ δ ψ δ ϖ λββ o the fith chai gives ϖ ρ > ϖ µ ξ µ ϖ ρ ϖ ρ ϖ δ ϖ δ τ λββ o the sixth chai gives τ µ > τ µ τ ρ τ ρ τ ρ τ δ τ δ ς λββ o the seveth chai gives ς µ > ς ρ ς µ ς ρ ς ρ ς δ ς δ λβ with λββ we have It is clea thatξ,ψ, ϖ eeset the sae euivalece class of fuzzy subgou hece will cout oce The fuzzy subgous eeseted by the fou: µ,ν, τ ad ς ae all distict, thus we have a total of five couts fo this eychai Siilaly fo othe cases 7 The Quateio gou Q 8 Q is foed by the uateios ±, ±i, ± j, ad ± 8 Q 8 8 The gou is geeated by i ad j with i, j i ad ji i j Its subgous ae, { },{, }, {,, i, i},{, j, j},{,,, } ad {, i, i, j, j,, } All the subgous ae oal ad cotai the subgou {, }, excet the tivial gou{ }

39 9 8 Maxial Chais fo Q 8 Thee ae thee axial chais fo this gou These ae: { } {, } {,, i, i} {,, i, i, j, j,, } { } {, } {,, j, j} {, i, i, j, j,, } { } {, } {,,, } {,, i, i, j, j,, } a Thee ae fou cooets fo each chai Theefoe a eychai βλα o the axial chai { } {, } {,, i, i} {,, i, i, i, i,, } subgou µ as follows: eesets a fuzzy β µ x λ α if if if if x x e x {, }\ { e} x {,, i, i} \ {, } {,, i, i, j, j,, } \ {,, i, i} Sice thee ae fou cooets i this chai, we have 5 distict fuzzy subgous o this chai, eeseted by the eychais λ λββ λ λβσ λλλ λβ λλ λλβ λ λβ λλ Usig this coutig techiue to deteie the ube of fuzzy subgous fo the etie gou, we obtai the followig table: Distict Keychais Nube of couts i all chais λ λλ λβ

40 λ λλλ λλβ λλ λββ λβα λβ λ Total Nube Thus Q 8 has distict fuzzy subgous 9 The gou Ζ fo A cyclic - gou is of the fo Maxial chais fo Ζ ad Ζ Ζ,, a ie Ζ Ζ, has oly oe axial chai of the fo { } ad if the cyclic gou Ζ, Ζ cotais the cyclic subgou of ode, we wite, fo a The case We have the chai { } I 9a ay fuzzy subgou of, λ, whee > λ > Let µ x ν x λ if if Ζ 9a if if x x x \ Ζ is euivalet to ay of the followig: { } x \ { }

41 if x ξ x, the if x \ { } µ, ν λ, ξ It is clea that µ ν, ξ because by costuctio > λ > µ fo x, y \ { } Now x µ y while ν x > ν y ad ξ x > ξ y fo the sae x ad y It is clea that µ is ot euivalet to ν ad ξ We also obseve that ν x > ν y ξ x > ξ y but the suν suξ, theefoe ν is ot euivalet to ξ Sice thee is oly oe chai, each eychai couts oce o the axial chai, esultig i thee distict euivalece classes of fuzzy subgous fo this gou b The case We have the axial chai { } with seve distict classes of fuzzy subgous viz, λ,, λλ, λβ, λ ad Fo the above it is clea that usig the euivalece stated i sectio µ ad ν λ ae ot euivalet as µ x µ y fo x \ { }, y \ but ν x > ν y fo the sae x ad y because by assetio > λ Now we obseve that 7 A siila aguet ca be used to show that the axial chai { } Ζ Ζ Ζ of the gou Ζ has 5 distict fuzzy subgous ad5 This suggests theoe Theoe: Fo ay o Ζ Ν thee ae Poof See Poositio [ 5 ] distict euivalece classes of fuzzy subgous

42 O the goug whee ad ae distict ies ad N Theoe: The ube of axial chais fo the gou G is fo Poof Staightfowad See illustatios, Figues, ad ude list of figues The ube of fuzzy subgous of the goug whee ad ae distict ies ad N I this sectio we wat to deteie a geeal foula fo the ube of distict fuzzy subgous fo the gou G whee ad ae distict ies also deived i[ 5 ] We advace a few values of to otivate theoe 8 Although a oof of the sae theoe was give by Muali ad Maaba i[ 5 ], we give a diffeet vesio of the oof as a way of illustatig how ou ethod of i-extesio is used 5 The case that is G Fo theoe with, G has axial chais ad these ae: { } Ο Ο { } Each axial chai has thee cooets, thus coesodig to each axial chai thee ae seve distict euivalece classes of fuzzy subgous give by the eychais,, λβ,, λ, λλ adλ If the two chais ae cosideed, we obtai a total of eleve o-euivalet fuzzy subgous as exlaied below: The eychais, λλ, each eesets the sae fuzzy subgou if both axial chais ae cosideed, thus givig a total of thee o-euivalet fuzzy subgous The eychais λ,, λβ adλ each behaves as a uiue fuzzy subgou with efeece to each axial chai, hece each couts twice givig a total of eight o-euivalet fuzzy subgous This gives a total of eleve o- euivalet fuzzy subgous fo the gou

43 Below we exlai how we aive at this ube of couts: Suose we tae fo exale the eychai, it gives a cout of oe i both chais because it is the sae fuzzy subgou i both cases, that is µ x, x Ζ Ζ We obseve that if we let µ λ ad ν λ fo the fist ad secod chais esectively, the µ x > µ y but y ν x { } ν > fo the sae x { } y, theefoe the sae eychai eesets diffeet euivalece classes of fuzzy subgous whe obseved i the cotext of each chai, thus the cout two A siila aguet holds fo the double cout of the est ad The case that is the gou Ζ Ζ Fo this gou, theefoe we have axial chais by Theoe ad these ae: { } { } { } { } { } { } { } Thee ae fou levels fo each chai Thus coesodig to the chai { } { } { } fo exale we have 5 distict euivalece classes of fuzzy subgous as listed below λ λββ λ λβα λλλ λβ λλ λλβ λ λβ λλ, whee > λ > β > α > Cosideig all the chais it ca be show usig this coutig techiue that thee ae distict euivalece classes of fuzzy subgous Rea: This is how the coutig techiue goes: fo exale the eychai couts oce i all the axial chais because it is the sae fuzzy subgou i all cases that is µ x, x Ζ Ζ The eychai λ couts twice if all chais ae cosideed because if we let µ λ, ν λ ad ξ λ be thee eychais coesodig to the fist,

44 secod ad thid chais esectively, they ae distict fuzzy subgous sice fo the sae x ad y Ο we have µ x > µ y but ν y < ν x ad ξ x < ξ y fo exale x,, y, I othe wods the eychai λ o the fist ad secod axial chais eeset the sae fuzzy subgou while it eesets a diffeet euivalece class o the thid axial chai Now usig this coutig techiue, we have the followig table which coletes the etie cout Distict Keychais Nube of couts i all chais λ λλ λβ λ λλλ λλβ λλ λββ λβα λβ λ Total Nube of Theefoe the gou G has distict fuzzy subgous We obseve that 8 7 The case whe that is Ζ Ζ Fo the gou G Ζ Ζ we have, thus we have axial chais fo this gou These ae:

45 5 { } { } a { } { } { } { } b { } { } { } c { } { } d Thee ae five levels fo each axial chai Coesodig to each axial chai we have distict fuzzy subgous, give by the eychais:, λ,, λλ, λβ, λ,, λλλ, λλβ, λλ, λββ, λβδ, λβ, λ,, λλλλ, λλλβ, λλλ, λλββ, λλβδ λλβ, λλ, λβββ, λββδ, λββ, λβδδ, λβδγ, λβδ, λβ, λ ad If all these distict fuzzy subgous ae tae idividually fo all the fou chais we get 79 o-euivalet fuzzy subgous fo the gou G We also obseve that This otivates theoe 8 Theoe: 8 The ube of distict fuzzy subgous fo the gou G is fo N Poof We ove by iductio o The foula holds fo, ad as show above Suose the stateet is tue fo, that is G has distict fuzzy subgous We ae goig to ae use of the lattice diaga of subgous of ad exted fo the two odes ad to the lattice diaga of subgous of The subgou is witte as o sily

46 We show that the theoe is tue fo The ube of fuzzy subgous of that ed with a ozeo i is oe oe tha those that ed with a zeo i Thus the ode subgou [ ] has o- euivalet fuzzy subgous edig with a ozeo i, ad thee ae fuzzy subgous edig with a zeo i Each of the foe yields thee distict fuzzy subgous i the subgou as follows: A eychai i is of the fo α α α Now fo α, we ca oly exted to α α α α, α α α β ad α α α eychais i fo < β < α Theefoe yields fuzzy subgous i ad because o zeo we ca oly attach a zeo The ode has eais the sae o-euivalet [ ] fuzzy subgous fo theoe Siilaly thee ae subgous that will give ise to ew fuzzy subgous whe alied to extesios Suose α α α is a eychai i obtai seve eychais viz: α with Extedig to α α α α α, α α α α β, α α α α, we α α α ββ, α α α βα, α α α β ad α α α fo < β < α ad < α < β Thee have bee couted befoe viz fuzzy α α α α α, α α α ββ, α α α, though Thus yields eychais i

47 7 Siilaly eychais i edig with zeo do ot cotibute ew fuzzy subgous as these have bee couted whe extedig fo to Suig u we get This coletes the oof Isoohic Classes of Fuzzy Subgous A atheatical object usually cosists of a set ad soe atheatical elatios ad oeatios defied o the set A collectio of atheatical objects that ae isoohic fo a isoohis class I defiig isoohis classes theefoe the oeties of the stuctue of the atheatical object ae studied ad the aes of the eleets of the set cosideed ae ielevat Defiitio: A isoohis class is a euivalece class fo the euivalece elatio defied o a gou by a isoohis We ae goig to use the defiitio of isoohis give i sectio The otio of euivalece is a secial case of fuzzy isoohis, that is if two fuzzy subgous ae euivalet the they ae isoohic but ot vice vesa Defiitio: Two o oe axial chais ae isoohic if thei legths ae eual ad the coesodig cooets ae isoohic subgous Nube of Isoohic classes fo selected fiite gous: The syetic gou S see sectio S has the followig axial chais a { e} B S i { e} B S ii { e} B S iii { e} B S iv

48 8 We obseve that chai i is ot isoohic to the othe chais ii ad iii which ae isoohic to each othe, theefoe will be viewed as distict fo othes But ii ad iii will be viewed as oe chai So calculatig the ube of isoohic classes of fuzzy subgous we obtai the followig i tabula fo: Distict Keychais λ λλ λβ λ Total ube of isoohic classes Nube of ways each Keychai couts Coets Fo the gou S we have fewe isoohic classes of fuzzy subgous tha euivalece classes The Quateio gou Q 8 This gou has the followig axial chais as eseted i chate thee { } {, } {,, i, i} {,, i, i, j, j,, } ** { } {, } {,, j, j} {, i, i, j, j,, } *** { } {, } {,,, } {,, i, i, j, j,, } **** **, *** ad **** ae all isoohic sice by costuctio i j they ae viewed as oe chai whe coutig the ube of isoohic classes I sectio we established that each chai has 5 o-euivalet fuzzy subgous that ca be eeseted by the followig sybols: λ λββ λ λβσ λλλ λβ λλ λλβ λ λβ λλ Sice all chais cout as oe, thee ae 5 isoohic classes of fuzzy subgous fo Q 8

49 9 Note: Thee ae fewe isoohic classes of fuzzy subgous tha euivalece classes The gou G This gou has the followig axial chais { } { } { } { } The two chais ae ot isoohic, thus each cotibutes to the ube of isoohic classes We established i chate thee that thee ae 7 distict fuzzy subgous fo each chai, these ae: λβ λ λλ λ Fist we eset a table of eychais ad the ube of isoohic classes eeseted by each eychai We cout these as i the case of euivalece classes ad obtai the followig table: Distict Keychais λ λλ λβ λ Total ube of isoohic classes Nube of ways each Keychai couts We obseve that the ube of euivalet fuzzy subgous is eual to the ube of isoohic classes fo this gou The goug Thee ae thee axial chais fo this gou as show below: { } { } i { } { } ii

50 { } { } { } 5 iii We obseve that chai iii cotais a cyclic subgou { }, theefoe is ot isoohic to eithe i ad ii Also i ad ii ae ot isoohic because ad ae diffeet ies Thus the ube of isoohic classes of fuzzy subgous is eual to the ube of euivalece classes fo this gou So fo the gou, the ube of isoohic classes of fuzzy subgous is eual to the ube of euivalece fuzzy subgous ad is give by the foula See theoe [ 5 ] Now if we ivestigate the gou G, we stat with fo exale which has the followig axial chais { } { } { } { } { }, they ae all isoohic thus has isoohic classes but has euivalece classes of fuzzy subgous I geeal G has oly oe subgous of odes ad All subgous of ode ae isoohic, hece all the axial chais ae isoohic which ilies that G has 7 isoohic classes of fuzzy subgous fo all ies Fo the gou G we have the followig axial chais { } { } { } { } { }, { } { } { }, The last two axial chais ae isoohic ad will be viewed as oe chai while the fist two ae also isoohic So the ube of isoohic classes of fuzzy subgous fo G is

51 5 Siilaly it ca be show that the gou G has 5 5 isoohic classes of fuzzy subgous

52 5 Chate Fou ON THE MAXIMAL CHAINS OF THE GROUPS G AND G s Itoductio Sice the cocet of axial chais lays a cucial ole i facilitatig the chaacteizatio of fuzzy subgous of aticula gous, i this sectio we wish to deteie a foula fo the ube of axial chais fo the gou G ad ossibly cojectue o the foula fo the gou G fo all values of s,, s ad fo all, ad distict iesto accolish this we fist begi with studyig the gou Ngcibi i [ ] G studied the classificatio of abelia gous of the fo G ad obtaied the followig esults which we ut dow i the fo of leas without oof Maxial Chais of G Lea: G has axial chais Poof [ ] Lea: G has axial chais Poof [ ] Lea: G has axial chais Poof [ ] Lea: G has axial chais Poof[ ]

53 5 We also iclude the followig defiitio that will be used i the decoositio of gous whe deteiig the ube of axial chais of these selected gous Defiitio: A axial subgou subgou ' G of a gou G is a oe subgou of G such that o oe " G of G stictly cotais ' G 5 Maxial Chais of G Sice ou ultiate goal is to establish the foula fo the ube of axial chais fo the gou G, we accolish this by fixig ad fo that aticula value of, values of ae advaced to idetify a atte Whe we have G ad advacig a few values of say,,, we obseve that hee ae axial chais see tee diagas of subgous fo,, ad Figues Oe, Two ad Thee ad by syety G has axial chais