THE ISOMORPHISM PROBLEM FOR CAYLEY GRAPHS ON THE GENERALIZED DICYCLIC GROUP

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1 IJAMM 4:1 (016) Mach 016 ISSN: Avalale at DOI: THE ISOMORPHISM PROBEM FOR CAYEY RAPHS ON THE ENERAIZED DICYCIC ROUP Pedo Manuel Domnguez Wade a and Alan Domnguez Fuentes a Depatment of Mathematcs, Matanzas Unnvesty, Cua Depatment of Infomatcs, Catholc Unvesty Pontfces of Ro de Janeo, Bazl Astact et Dc ( a, ) e the genealzed dcyclc goup. In ths pape, we show that f Dc ( a, ) has ode k s, k > 1, k N + 1, s 1 o, >, goup wth espect to gaphs. then Dc ( a, ) s a Cayley somophc Keywods: Cayley gaph, CI-gaph, CI-goup, genealzed dcyclc goup. * Coespondng autho. E-mal addess: pedoalgealneal@gmal.com (Pedo Manuel Domnguez Wade). Copyght 016 Scentfc Advances Pulshes 010 Mathematcs Suect Classfcaton: Pmay 05C5; Seconday 0B5. Sumtted y Zek Kasap. Receved Januay 5, 016

2 0 P. M. D. Wade and A. D. Fuentes / IJAMM 4:1 (016) Intoducton In 1967, Ádám [1] conectued that any two Cayley gaphs of Z m ae somophc f and only f they ae somophc y an automophsm of Z m. A counte example to ths conectue was quckly found n 1970 y Elspas and Tune [6]. Posteoly, Muzychuk completed the polem of detemnng whch values of m have the popety that any two Cayley gaphs of Z m ae somophc f and only f they ae somophc y an automophsm of Z m, povng that f m s squae fee, then Z m [8] and Z m [9] have ths popety. The only othe values of m wth ths popety ae 8 and 9 []. Ádáms conectue was quckly genealzed to the followng polem. Polem 1.1. Fo whch fnte goups s t tue that any two Cayley gaphs of ae somophc f and only f they ae somophc y an automophsm of? A fnte goup wth ths popety wll e called a Cayley somophc goup (fo evty CI-goup ) wth espect to gaphs. Ths polem ae studed y many authos. odsl [7] poved that Z p s a CI-goup wth espect to gaphs fo p a pme. Baa [3] showed that the nonaelan goup of ode p s a CI-goup wth espect to gaphs, and the autho [4] showed that the nonaelan goup of ode pq, p and q dstnct pmes, s a CI-goup wth espect to gaphs f and only f q = o 3. Whle some othe esults on the aove polem ae known, othe than the aove mentoned esults of Muzychuk thee ae no known CI-goups wth espect to gaphs whee the ode of the goup has moe than thee pme factos. In ths pape, we show that f the genealzed dcyclc goup Dc ( a, ) has ode k s, k > 1, k N + 1, s 1 o, >, then Dc ( a, ) s a Cayley somophc goup wth espect to gaphs.

3 THE ISOMORPHISM PROBEM FOR CAYEY / IJAMM 4:1 (016) Pelmnay Results Defnton.. et e a fnte goup and let S e a suset of whch s closed unde nveson and does not contan the dentty. Then, the gaph Γ = Γ( ; S) gven y V ( Γ ) = and E( Γ) = {(, h) : h S} s called Cayley gaph. Defnton.3. et e a fnte goup. Fo g, defne f g : y f g ( h) = gh. Then = { f : g } s tself a goup, the left egula epesentaton of and s somophc to. g Remak.4. Clealy f Γ (, S) s a Cayley gaph on, then h h S and ( gh) gh S ae equvalent statements, so that f g Aut ( Γ), whee Aut ( Γ) s the automophsm goup of Γ. Theefoe, Aut( Γ). We make some comments aout of the nomalze N Aut( Γ )( ). It well known that fo the nomalze of n S, we have Thus, we have NS ( ) = Aut( ). N Aut( Γ)( ) = ( Aut( )) Aut( Γ) = ( Aut( ) Aut( Γ)) (.1) = Aut(, S), whee Aut (, S) : = { α Aut( ) αs = S}. et H e a sugoup of. Fom (.1), we deduce that N Aut( Γ )( H ) = N ( H ) Aut(, H, S), (.) whee Aut (, H, S) : = { α Aut(, S) αhα = H }.

4 P. M. D. Wade and A. D. Fuentes / IJAMM 4:1 (016) Defnton.5. et e the fnte goup and let Γ e some Cayley gaph of. We shall say that Γ s a Cayley somophc gaph (fo evty CI-gaph) of f gven any Cayley gaph Γ of such that Γ s somophc to Γ, then Γ and Γ ae somophc y some α Aut( ). The followng chaactezaton of the CI-gaph was poven y Baa and wll e used n ths pape. emma.6 (Baa [3]). Fo a Cayley gaph Γ of a fnte goup, the followng ae equvalent: (1) Γ s a CI-gaph of ; () gven a pemutaton φ S such that φ φ Aut( Γ), φ φ ae conugate n Aut ( Γ). and Othe chaactezaton of CI-gaph we may fnd n the followng lemma: emma.7. Fo a Cayley gaph Γ of a fnte goup, the followng ae equvalent: (1) Γ s a CI-gaph of ; h () gven a pemutaton φ S such that φh Aut( Γ) fo some N S ( ). Poof. Assume that Γ s a CI-gaph of. Accodng to emma.6, gven a pemutaton φ S such that φ φ Aut( Γ), and conugate n Aut ( Γ),.e., thee exsts g Aut( Γ) such that Fom (.3), we otan φ φ ae g g = φφ. (.3) φ g g φ =. (.4) 1 Theefoe, we may asset that φ g = h N S ( ). The convese mplcaton follows applyng newly the last lemma. So we ae done.

5 THE ISOMORPHISM PROBEM FOR CAYEY / IJAMM 4:1 (016) Theefoe, comnng the emmas.6 and.7 we otan: Poposton.8. Fo a Cayley gaph Γ of a fnte goup, the followng ae equvalent: (1) Γ s a CI-gaph of ; () gven a pemutaton φ S such that φ φ Aut( Γ), φ φ ae conugate n Aut ( Γ); and (3) gven a pemutaton φ S such that φh Aut( Γ) fo some h N S ( ). et R e a feld of chaactestc p o a complete dscete valuaton ng wth esdue feld of chaactestc p. We ecall that a mnmal p-sugoup Q of elatve to whch the ndecomposale R-module U s poectve s called a vetex of U, and t s defned up to conugacy n. Moeove, an RQ-module Z fo whch U s a summand of Ind Q ( Z ) s called a souce of U, and gven the vetex Q t s defned up to conugacy y elements of N ( Z ). If Z = R we say that U has tval souce. The followng lemma s well known. emma.9. A vetex of the tval R-module R s a Sylow p-sugoup of. The followng lemma wll e used n ths pape. emma.10. et e a fnte goup and let P e a p-sugoup of. If P Syl ( N ( P )), then P Syl ( ). p p Poof. Accodng to the last lemma, the p-sugoup P s a vetex of the tval RN ( P )-module. Snce the een coespondent of the tval RN ( P )-module s the tval R-module, applyng agan the emma.9 the esult follows.

6 4 P. M. D. Wade and A. D. Fuentes / IJAMM 4:1 (016) Some Popetes of the oup Dc ( a, ) k ls et Dc( a, ) = a, : a =, a = a, a = = 1 u dk dls e the fnte goup, whee k, s, and u ae nteges wth k > 1 and s 1. The postve ntege d s a dvso of u 1 and l s the multplcatve ode of u modulo d k. The goup s called genealzed dcyclc goup. et = lsq, 0 + < ls. Oseve that fo all element g = a ( 0 dk, 0 dls 1) we have g = a = a = u a = u lsq+ a = u + kq a. Thus, fo evey element g Dc( a, ), we may wte g = a ( 0 dk 1, 0 ls 1). Theefoe, we may asset that Dc( a, ) = dkls. Remak Oseve that when u = and s = 1, the goup s dhedal o geneal quatenon goup accodng to d = 1 o d =. Cente of the oup We denote the cente of the goup y Z ( Dc( a, ) ). et d e the u 1 geatest common dvso of k and. d In [5] was poved that k β lδ Z( Dc( a, ) ) = g Dc( a, ) : g = a d, whee β = 0,, dd 1, δ = 0,, s 1. Hence, we have Z( Dc( a, ) ) = dd s. (3.1) Remak 3.1. et I nn ( Dc( a, ) ) e nne automophsms goup. Then, k fom (3.1), we have I nn ( Dc( a, ) ) = k l, k =. d

7 THE ISOMORPHISM PROBEM FOR CAYEY / IJAMM 4:1 (016) Poposton et Dc ( a, ) e the genealzed dcyclc goup. Then fo the conugacy classes a gcd (, dls) = 1), we have Dc ( a, ) ( gcd(, dk ) = 1) and Dc( a, ) ( 1 < ls, a Dc ( a, ) Dc( a, ) k u 1 l and ( = = k k =, d = gcd( k, ). d d Poof. et we have a wth ( gcd (, dk ) = 1). Then fo any element a Dc( a, ) x y x y x y y u ( a ) a ( a ) = a. Snce 0 y l the esult follows. Futhemoe, x y x y x ( ) ( ) ( 1 u) a a = a. 1 u We may asset that ( 1 u) x = dd u x( u = ) y assumpton. So we dd ae done. emma et Dc ( a, ) e the genealzed dcyclc goup. (1) If gcd ( d, k ) = 1, then Aut( Dc( a, ) ) = dk ϕ( dk ) ϕ( dls), whee ϕ s the Eule s ph functon. () If gcd( d, k ) = d, then Aut( Dc( a, ) ) = dk ϕ( dk ) ϕ( ls). Poof. (1) We clam that n such case Dc( a, ) ( { 1,, dls 1}, gcd (, dls) = 1. Moeove, we may see that any α Aut ( Dc( a, ) ) s gven y α( a) = a, gcd(, dk ) = 1 x and α( ) = a, x = 0,, dk 1; gcd (, dls) = 1. Assume that A = { a : gcd(, dk ) = 1} and B = x { a : x = 0,, dk 1; gcd(, dls) = 1}. Thus, we may wte A = ϕ( dk ) and B = dkϕ( dls). Snce α ( a) = a A and ( ) a x β = B the esult follows.

8 6 P. M. D. Wade and A. D. Fuentes / IJAMM 4:1 (016) () Hee Dc( a, ) f and only f ( { 1,, ls 1} ), gcd(, dls) = 1. Thus, poceedng as n the case (1) the asseton follows. emma et Dc ( a, ) e the genealzed dcyclc goup. Then Aut ( Dc( a, ) ) s a solvale goup. Poof. We check two cases. Case I: gcd ( k, d) = 1 Accodng to the last lemma, we have Aut( Dc( a, ) ) = dk ϕ( dk ) ϕ( dls). et H e a sugoup of Aut ( Dc( a, ) ) gven y H = α Aut( Dc( a, ) ) α( a) = a, α( ) = ( gcd(, dls) = 1, 1 dls 1. We clam that H s aelan goup of ode ϕ ( dls). We consde the sugoup H = β Aut( Dc( a, ) ) β( a) = a ( gcd(, dk ) = 1, 1), β( ) = ( gcd(, dlk ) = 1), Dc( a, a a ) o Dc( a, ). We may asset that H 1 s also aelan dϕ( dk ) goup whose ode s gven y q, q { 1,, l}. Hence l I nn ( Dc( a, ) ) H1 s nomal sugoup of Aut ( Dc( a, ) ) of ode dkϕ ( dk ). Thus, we may wte Aut( Dc( a, ) ) = Inn ( Dc( a, ) ) H1 H. (3.) Snce I nn ( Dc( a, ) ), H1 and H ae solvale goups, fom (3.) the esult follows. Case II: gcd( k, d ) = d 1 Poceedng as n the last case the asseton follows. 4. Man Results Theoem et Dc ( a, ) e the genealzed dcyclc goup, whee d = 1, l =, s 1 and k > 1 s an odd nume. Assume that Γ ( Dc ( a, ), S) s a Cayley gaph on Dc ( a, ) and that denote the

9 THE ISOMORPHISM PROBEM FOR CAYEY / IJAMM 4:1 (016) hghest powe of dvdng f. Then f a f s the lagest nomal π -sugoup of a Hall π-sugoup n Aut ( Γ), eng π the set of all the odd pme dvsos of k and s. Poof. et us wte fo Aut ( Γ) and we wte H fo f f. Fstly, we oseve that f Z( Dc( a, ) ). Assume now that P H s a Sylow p-sugoup. Snce P s a nomal Sylow p-sugoup of Dc ( a ), fom (.), t follows that a, N ( P ) = N ( Dc( a, ) ) = Dc( a, ) Aut( Dc( a, ), S). (4.1) By emma 3.15, t follows that Aut ( Dc( a, ) ) s a solvale goup, so Aut ( Dc( a, ), S) s also solvale. Snce Dc ( a, ) s solvale, fom (4.1), we deduce that N ( Dc( a, ) ) s solvale. Thus, the Hall π -sugoups of N ( Dc( a, ) ) thee exst. et H e a fxed Hall π -sugoup of Aut ( Dc( a, ), S) and let P e a Sylow p-sugoup of H. Then we may asset that the semdect poduct N ( P ). Hence, we may wte Q = P P s a Sylow p-sugoup of N ( Q) = P N, whee N s the nomalze of P n Aut ( Dc( a, ), S). Theefoe Q Syl ( N ( Q)), so we may asset that Q s a Sylow p p-sugoup of y emma.10. Snce the last statement hold fo evey Sylow p -sugoup ( p π, = 1,, π ) of H H, we deduce that H H s Hall π -sugoup of, whch s what we need to pove. We now gve ou man esults n ths pape. Theoem et Dc ( a, ) e the dcyclc genealzed goup, whee d = 1, s 1, u = k 1 and k > 1 s an odd nume. Then Dc ( a, ) s CI-goup wth espect to gaphs.

10 8 P. M. D. Wade and A. D. Fuentes / IJAMM 4:1 (016) Poof. Assume that Γ s any Cayley gaph on Dc ( a, ). et us wte fo Aut ( Γ). We wll show that Γ s a CI-gaph. Suppose that φdc( a, ) φ, whee φ S Dc ( a, ). Accodng to Theoem 4.16, we may asset that f a f and φ f a f φ ae lagest nomal π -sugoup of a Hall π -sugoup of espectve, whee π s the set of all the odd pme dvsos of k and s. It well known that the Hall π -sugoups of a fnte goup (f they exst) ae congugacy, so we may asset that fo, some g, we have g fa f g = φ f a x { 1,, k} holds. Hence, we have f x ( a ) φ, gcd( k, ) = gcd( s, ) = 1, x gag = φa φ and g g = φ( a ) φ. (4.) Fom (4.), t follows that φ g = a φ ga. Theefoe, we have ( φ g ) = ( a ( φ g ) a )( a ( φ g ) a ). (4.3) Thus, fom (4.3), we otan the followng equalty: Hence, fom (4.4), we otan + 1 ( ) ( ) ( + 1 φ g a φ g = a ). (4.4) ( φ g ) a ( φ g ) = a. (4.5) Applyng (4.) secund pat, we deduce that the equalty (4.5) s tue f and 1 only f φ g f, so we may asset that φ. Theefoe, accodng to emma.6, the Cayley gaph Γ s CI-gaph. So we ae done. Theoem et Dc ( a, ) e the dcyclc genealzed goup wth Dc ( a, ) =, >. Then Dc ( a, ) s CI-goup wth espect to gaphs.

11 THE ISOMORPHISM PROBEM FOR CAYEY / IJAMM 4:1 (016) Poof. et us wte fo Aut ( Γ). Assume that Γ s a Cayley gaph on Dc ( a, ). We wll pove that Γ s CI-gaph. By assumpton, we clam that Aut ( Dc( a, ) ) s a -goup. Thus, we may asset that Aut ( Dc( a, ), S) s a -sugoup of Aut ( Dc( a, ) ). Hence, fom (.1), we deduce that the nomalze N ( Dc( a, ) ) s a -goup. We wll pove that N ( Dc( a, ) ) s a Sylow -sugoup of. et R e a feld of chaactestc. Then we may asset that RAut ( Dc( a, ), S) s an ndecomposale RN ( Dc( a, ) )-module wth vetex Dc ( a, ) and tval souce. Assume that P s a Sylow -sugoup of such that N ( Dc( a, ) ) P. Theefoe, snce RP s ndecomposale applyng the een coespondence, we deduce the followng holds: P = N ( Dc( a ) )., Suppose that φdc( a, ) φ, whee φ S Dc ( a, ). We may asset that N ( φdc( a, ) φ ) Syl ( ). p Snce all the Sylow -sugoups ae conugacy n t follows that Dc ( a, ) and φdc ( a, ) φ ae conugacy. Thus, applyng the emma.6, we conclude that Γ s a CI-gaph, whch s what we need to pove. Remak It s well known that the quatenon goup s CI-goup wth espect to gaphs. Oseve that the quatenon goup s a patcula case of Dc ( a, ) unde the condtons of the last theoem. Refeences [1] A. Ádám, Reseach polem -10, J. Comn. Theoy (1967), 393. [] B. Alspach and T. D. Pasons, Isomophsm of cculant gaphs and dgaphs, Dscete Math. 5() (1979), [3]. Baa, Isomophsm polem fo a class of pont-symmetc stuctues, Acta Mathematca Academae Scentaum Hungacae 9(3) (1977),

12 30 P. M. D. Wade and A. D. Fuentes / IJAMM 4:1 (016) [4] E. Doson, Isomophsm polem fo Cayley gaphs of 3 Z p, Dscete Math. 147(1-3) (1995), [5] P. Domínguez Wade, Modula epesentatons of the goup MQ ove the ng K M, Asan Jounal of Mathematcs, Intenatonal Pess 10(4) (006), [6] B. Elspas and J. Tune, aphs wth cculant adacency matces, J. Comn. Theoy Se. B 9 (1970), [7] C. D. odsl, On Cayley gaph somophsms, As Comn. 15 (1983), [8] M. Muzychuk, Ádám s conectue s tue n the squae-fee case, J. Comn. Theoy Se. A 7 (1995), [9] M. Muzychuk, On Ádám s conectue fo cculant gaphs, Dscete Math. 176 (1997), [10] T. Okuyama, Module coespondence n fnte goups, Hokkado Mathematcal Jounal 10 (1981), [11]. R. Ronson and R. Staszewsk, On the epesentaton theoy of π - sepaale goups, J. Algea 119(1) (1988), 6-3. g