Some Properties of The Normalizer of o (N) on Graphs

Transcription

1 Jornal of te Applied Matematic Statitic and Informatic (JAMSI) 4 (008) o Some Propertie of Te ormalizer of o () on Grap BAHADIR O GULER AD SERKA KADER Abtract In ti paper we give ome propertie of te normalizer ℵ of o() in PSL( ) and ow tat te nmber of edge are related to te period of grop element Matematic Sbject Claification: F06 0H05 Keyword: Te normalizer of o() imprimitive action borbital grap ITRODUCTO Let PSL( ) denote te grop of all linear fractional tranformation T : z az + b cz + d were abcd are real and ad bc = For any natral nmber o() denote te bgrop of PSL( ) wit integral coefficient and property tat c 0 mod We all in te eqel freely e x matrice to repreent tranformation o tat T i repreented by te pair of te matrice a b ± (abcd ad bc = ) () c d We all write tee matrice a eqal were convenient in matrix calclation Following [] we denote by ℵ te normalizer of o() in PSL( ) coniting of all te matrice ae c b de were all letter are integer e and ( = ()) i te larget divior of 4 for wic wit te ndertanding tat te determinant of te matrix i e > 0 and tat r mean tat r and (r r ) = ( r i called an exact divior of ) We now define 0 ( / : ) Γ to be te et 77

2 BOGULER SKADER a b ad ( bc) = c d wic i a bgrop of PSL( ) It can be eaily een tat ti grop i generated by o() and 0 THE ACTIO OF ℵ O ˆ Te element x y in ˆ will be in redced form tat i ( xy ) = Since x = x ti y y a b repreentation i not niqe Te action of te matrix c d on x y i a c b d : x y ax + by cx + dy and it i te ame on x y α α α3 αn THEOREM [] Let be any integer and = 3 p3 p n te prime power decompoition of Ten ℵ i tranitive on ˆ if and only if α 7 α 3 and α for i = 3 n From now on will be a in Teorem i LEMMA Let be a above Ten te tabilizer of a point of ˆ i an infinite cyclic grop of ℵ PROOF Since te action i tranitive te tabilizer of any two point in ˆ are conjgate in ℵ So it i fficient to conider te tabilizer ℵ of Ti i eaily een 78

3 SOME PROPERTIES OF THE ORMALIZER OF O() O GRAPHS b to conit of te element of te form and terefore 0 grop generated by 0 ℵ i te infinite cyclic We conider te imprimitivity of te action of ℵ on ˆ a follow: Generaly let (G Ω) be a tranitive permtation grop coniting of a grop G acting on a et Ω tranitively We will call (G Ω) imprimitive if Ω admit ome G-invariant eqivalence relation different from (i) te identity relation α β if and only if α = β ; ii) te niveral relation α β for all α β Ω Oterwie (G Ω) i called primitive Te eqivalence clae are called block LEMMA 3 [3] Let (G Ω) be tranitive Ten (G Ω) i primitive if and only if G α te tabilizer of a point α i a maximal bgrop of G for eac α Ω According to te above lemma if for ome α in Ω G H G α following relation on Ω make (G Ω) imprimitive: ten te g( α) g ( α) if and only if g g H () ow a a pecial cae ince ℵ o ( /:) ℵ we apply tee idea to te cae were G i te normalizer ℵ Ω i ˆ and G α i ℵ In ti cae if we find integer b and d c tat te element 0 ( re ) e b g = 0 de r v = ˆ ten i in ℵ wit determinant 0 e and ending to r were ( ) ( ) 0 e = = e e = e = e 0 e and e = e e x x Likewie given w = in ˆ tere exit g ℵ c tat g ( ) = Terefore by y y () 79

4 BOGULER SKADER v w= g( ) g ( ) ℵ rye x f 0 mod were y f ( for x y ) are defined a e ( for r it can be een if ) repectively After ome calclation r = 0 tat e = and y 0 mod And ere y 0 mod ow tat y 0 mod Coneqently we ave te block x [ ]: ˆ = y 0 mod () y From te imprimitivity te nmber of block nder i te index ℵ: ( /: ) o wic i r were i a above r i te nmber of ditinct prime dividing and = a in [] a follow 3 3 α α ( ) ( ) = 9 = 4 = 3 oterwie oterwie 3 SUBORBITAL GRAPHS FOR ℵ O ˆ Since (ℵ ˆ ) i a tranitive permtation grop ten ( ℵ ˆ ) i a permtation grop wit te action: for ˆ ( α β) and g ℵ g( α β ): ( g( α ) g( β )) = Te orbit of ti action are called borbital of ℵ From te orbit O ( α β ) we from a borbital grap ( α β ) : it vertice are te element of ˆ and tere i a directed edge from a to b if ( ab ) O ( α β ) If O ( α β ) = O( β α ) ten te grap conit of pair of oppoitely directed edge; we will replace c edge by ndirected edge In ti cae ve ave an ndirected grap wic will be called elf-paired If O ( α β ) O( β α ) ten ( β α ) i jt ( α β ) wit te arrow revered In ti cae we call ( α β ) and ( β α) paired grap Since ℵ i tranitive on ˆ eac borbital contain a pair ( v) for ome v ; writing 80

5 SOME PROPERTIES OF THE ORMALIZER OF O() O GRAPHS v = wit n > 0 and (n) = we denote te borbital O ( ) by O n and te n n grap by n Tee idea were firt introdced by Sim [8] and are alo decribed in a paper by emann [7] and in book by Tzk [9] and by Bigg and Wite [4] te empai being on application to finite grop Uing Teorem 4 in [] we ee tat tere exit integer m c tat O ( ) = n O ( ) Terefore we can drive te following m r x THEOREM 3 i an edge in m if and only if tere exit ome e y wit e e ten eiter (a) ry x = m e and x r mod ( m ) y mod m e or (b) ry x = m e and x r mod ( m ) y mod m e PROOF Let r x i an edge in y m Ten tere i ome element ae b A = c de in ℵ of determinant e ending to r and m to x y and terefore ae r = c and ( ae + ( bm) ) = ( c + dem ) x Since te determinant of te matrix A i y e and tat ( 0 ) = ten a = r and = c e Tat i e tat x =± ( a + ( bm) ) y ( c dm) e =± e + From ti we get Likewie we can ee 8

6 BOGULER SKADER ae b ( bm) i j ae ae + ( ) er ( ) ex = i j c de 0 m c c = (3) dem ( ) e ( ) ey + were i j = 0 If i = j = 0 ten ( m ) x r mod y mod m e and taking determinant of (3) we ee tat ry x = m e Similarly if i = j = 0 (or i = 0 j = ) we obtain (b) Converely if (a) old wit te given condition ten exit integer b and d c tat x = r+ ( bm ) and y = + dm It can be eaily own tat te e element re e follow imilarly b i in ℵ and end to r de and m to x y From now on for te ake of implicity ppoe tat m i If (b) old te proof THEOREM 3 If v - mod ten te borbital grap and v are paired PROOF Sppoe tat v mod and r x y i an edge in Ten ing Teorem 3 tere exit ome integer tat e e and (a) or (b) i atified Sppoe (a) old Ten ry x = e and x r mod ( ) y mod e Since v mod ten r vx mod e vy mod Ten x y r i an edge tat i and in v v are paired COROLLARY 33 i elf-paired if and only if mod PROOF Sppoe O ( ) = O ( ) Ten tere exit ome ae b ϕ = in ℵ c tat ending to c de and to From ti ϕ mt be e b and ten e i eqal to So mod e e 8

7 SOME PROPERTIES OF THE ORMALIZER OF O() O GRAPHS Converely Terefore e mod ten exit ome integer b c tat b i in ℵ and atified deired condition = + b 4 THE GRAPH F We let F be te bgrap of woe vertice form te block [ ] So by Teorem 3 we ave THEOREM 4 r x y i an edge in F if and only if eiter (a) x r mod and ry x = or (b) x r mod and ry x = An atomorpim of te grap F i a permtation of [ ] wic take edge to edge In view of ti it can eaily een tat Γ o ( / : ) < At F THEOREM 4 Γ ( / : ) o permte te vertice and te edge of tranitively F PROOF We do calclation only for edge Sppoe a b c and d k m t are two edge in F Ten tere exit T 0 T in Γ ( / : ) o c tat T 0 a c 0 b d and T k Terefore te element 0 m t T : = T o T i a reqired tranformation in Γ ( / : ) o - 0 DEFIITIO 43 By a directed circit in we mean a finite eqence v v v m of different vertice c tat v v v m v were m 3 anti-directed circit denote te configration like te above wit at leat one arrow (not all) revered If m = 3 ten te circit i called a triangle And more we call te configration v v v a elf-paired edge ; an 83

8 THEOREM 44 (a) Te grap F BOGULER SKADER contain directed triangle if and only if ± + 0 mod (b) F > contain no anti-directed triangle PROOF (a) Let a c k a be a directed triangle in F By Teorem 4 b d b we may ame tat te triangle a te form 0 x y 0 Since x y ten x mod 0 and -y = - tat i y = From x we get tat eiter mod x and x= or x mod and x = Terefore x = or x = + Ten te triangle i From te triangle we obtain ± + 0 mod ow ppoe tat ± + 0 mod Ten by Teorem 4 te circit i a directed triangle in F (b) By Teorem 4 witot lo of generality we may ame tat we ave an antidirected triangle a Ten eiter b mod b = or b mod b = On te oter and b give b mod So if b= + ten ( + ) mod and if b= ten ( ) mod Terefore we get 0mod a contradiction COROLLARY 45 Te bgrop Γ o ( / : ) of ℵ doe not contain elliptic element of order 4 or 6 PROOF Ti follow from Teorem in [3] 84

9 SOME PROPERTIES OF THE ORMALIZER OF O() O GRAPHS COROLLARY 46 Γ ( / : ) o contain an elliptic element of order 3 if and only if tere exit in te grop U of nit mod c tat F contain a triangle PROOF Sppoe tat a b ϕ = i an elliptic element in c Γ ( / : ) d o of order 3 Ten a+ d = ± and ad mod And a( ± a) mod tat i a ± a+ 0 mod A ( a ) = ( a ) = ten tere exit in U c tat triangle a mod So ± + 0 mod By Teorem 44 F contain a Converely ppoe tat F contain a triangle By Teorem 44 we get te triangle From ti we obtain order 3 ± + φ = ± in Γ o ( / : ) of COROLLARY 47 Let or 3 3 Ten F Tat i Γ ( / : ) o doe not ave elliptic element of order 3 doe not contain any triangle PROOF Sppoe firt tat Ten So ± + 0 mod tat i ± + 0 mod Hence F doe not contain any triangle If 3 3 ten 3 Since follow ± + 0 mod 3 ± + 0 mod Terefore te relt COROLLARY 48 Let a c e a be a triangle in F b d f b Sppoe tat 0 mod ± + bt ± + 0 mod Ten tere exit one and only one 85

10 BOGULER SKADER elliptic element T in Γo( / : )\ Γ o( ) of order 3 c tat a c c e e a T = T= T= b d d f f b PROOF By Teorem 4 tere exit ome T 0 in Γ o ( / : ) ending te given triangle to te triangle ± + ten U = ± i an elliptic element in Γo( / : )\ Γ o( ) of order 3 So te element T : = T ouo T o o i an elliptic element of order 3 If T i in Γ ( ) o ten U i in T o Γ ( )ot o o = Γ o o ( ) contradiction It i obvio tat T atifie deired condition Te convere of te above corollary i alo tre Tat i a COROLLARY 49 Let a b ϕ = c d be an elliptic element of order 3 in Γo( / : )\ Γ o( ) Ten tere exit a niqe in U wit a mod c tat ± + 0 mod and ± + 0 mod and i a triangle in F PROOF Since in an elliptic element of order 3 and i in Γo( / : )\ Γo( ) ten by Corollary 47 and 3 and o i jt 3 It may appen tat bc Ti doe not occr becae in tat cae from te determinant we get ad mod tat i a( a± ) mod or a a 0 mod ± + So a ± a+ 0 mod 3 a contradiction Terefore bc Hence we get a ± a+ 0 mod bt a a 0 mod ± + Since (a ) = (a ) = tere i ome in U wit a mod Terefore 0 mod ± + Coneqently we get te triangle a 86

11 REFERECES SOME PROPERTIES OF THE ORMALIZER OF O() O GRAPHS Akba M 989 PD Tei Univerity of Sotampton Akba M and Bakan T 996 Sborbital grap for te ormalizer of Γ ( ) o Trki J Of Matematic Akba M and Singerman D 99 Te ignatre of te ormalizer of Γ ( ) o London Mat Soc Lectre ote 65 CUP Bigg L and Wite AT 979 Permtation grop and combinatorial Strctre London Mat Soc Lectre ote Serie 33 Cambridge Univerity Pre Cambridge 5 Jone GA Singerman D and Wick K 99 Te modlar grop and generalized Farey grap London Mat Soc Lectre ote Serie 60 CUP Le Veqe W J 977 Fndamental of mber Teory Addion-Weley Reading Ma 7 emann P M 977 Finite permtation grop edge-colored grap and matrice Topic in grop teory and comptation Ed M P J Crran Academic Pre London ew York San Francico 8 Sim C C 967 Grap and finite permtation grop Mat Z Tzk T 98 Finite Grop and finite geometrie Cambridge Univerity Pre Cambridge Baadır O GULER Department of Matematic Te Univerity of Rize 5300 Rize-Trkey; Serkan KADER Department of Matematic Te Univerity of Karadeniz Tecnical 6080 Trabzon-Trkey Received September