K-NACCI SEQUENCES IN MILLER S GENERALIZATION OF POLYHEDRAL GROUPS * for n

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1 Iraa Joral of See & Teholog Trasato A Vol No A Prted the Islam Rebl of Ira Shraz Uverst K-NACCI SEQUENCES IN MILLER S ENERALIZATION OF POLYHEDRAL ROUPS * O DEVECI ** AND E KARADUMAN Deartmet of Mathemats Falt of See ad Letters Kafas Uverst Tre Atatr Uverst Deartmet of Mathemats Falt of See Erzrm Tre Emal: odeve6@hotmalom edma@ataedtr Abstrat A -a seee a fte gro s a seee of gro elemets whh gve a tal (seed) set eah elemet s defed b for for for I ths aer we eame the erods of the -a seees Mller s geeralzato of the olhedral gros ; ; ; ; for a Kewords K-a seee erod dhedral gro olhedral gro INTRODUCTION The std of Fboa seees gros bega wth the earler wor of Wall [] where he osdered Fboa seees of the l gros C Wlo eteded the roblem to abela gros [] I [] the Fboa legth of a -geerator gro s defed The oet of Fboa legth for more tha two geerators has bee osdered [] ad [] Prolf o-oerato of Cambell Dooste ad Robertso eaded the theor to some fte smle gros [] The theor has bee geeralzed [6] [7] to the ordar -ste Fboa seees fte lotet gros The t s show [8] that the erod of - ste geeral Fboa seee s eal to the legth of the fdametal erod of the -ste geeral rerree ostrted b two geeratg elemets of the gro of eoet ad lote lass Karadma ad Yavz showed that the erods of the -ste Fboa rerrees fte lotet gros of lote lass ad a rme eoet are ( ) for 97 where s rme ad ( ) s the erods of ordar -ste Fboa seees [9] The -ste geeral Fboa seees fte lotet gros of lote lass ad eoet ad the -ste Fboa seees fte lotet gros of lote lass ad eoet are dsssed [] ad [] resetvel I [] the relatosh betwee a mber of rerree sms volved the th term of the last omoet of the Fboa seees fte lotet gros of lote lass ad eoet ad the oeffets of the bomal formla has bee vestgated Ko roved that erods of the -a (ste Fboa) seees the dhedral gro were eal to [] Other wor o Fboa legth s dsssed [] ad [] Reetl the wors have bee doe o the -a seees [6-8] Reeved b the edtor Febrar 9 ad fal revsed form Deember Corresodg athor

2 76 O Deve / E Karadma Ths aer s related to the erods of the -a seees Mller s geeralzato of the olhedral gros ; ; ; ; for a Defto A -a seee a fte gro s a seee of gro elemets for whh gve a tal (seed) set eah elemet s defed b for for We also rere that the tal elemets of the seee geerate the gro ths forg the -a seee to reflet the strtre of the gro It s mortat to ote that the Fboa legth of a gro deeds o the hose geeratg -tle The -a seee of a gro geerated b s deoted b F( ; ) ad ts erod s deoted b P( ; ) Defto For a ftel geerated gro A A a a a the seee a s alled the Fboa orbt of wth reset to the geeratg set A deoted F A Note that the orbt of a -geerated gro s a - a seee -ste Fboa seee the tegers modlo m a be wrtte as F ( Z m;) A -ste Fboa seee of a gro of elemets s alled a Fboa seee of a fte gro A fte gro s -a seeeable f there ests a -a seee of sh that ever elemet of the gro aears the seee A seee of gro elemets s erod f after a erta ot t ossts ol of reettos of a fed sbseee The mber of elemets the reeatg sbseee s alled erod of the seee For eamle the seee abdebdebde s erod after the tal elemet a ad has erod A seee of gro elemets s sml erod wth erod f the frst elemets the seee form a reeatg sbseee For eamle the seee abde f abde f abde f s sml erod wth erod 6 where Remar The olhedral gro l m for lm> s defed b the resetato or The olhedral gro l m z : z z l m z : lm s fte f ad ol f the mber lm m l lm lm s ostve ad the order of l m beg lm l m T l m These gros are also alled tragle gros ad are deoted b Remar Mller s geeralzato of the olhedral gro resetato l m for lm> s defed b the l m : Iraa Joral of See & Teholog Tras A Volme Nmber A Atm

3 K-a seees mller s Its order s that of l m mltled b the erod of etral elemet S l m 77 If ths erod s fte a dvsor elds a fator gro l m ; m l ; defed b S l m : S For more formato o these gros see [9] MAIN RESULTS AND PROOFS Theorem Let be the gro defed b the resetato : S S get We P ( ) Proof: We frst ote that If the seee wll be as follows: If P ; bease of If P ; 6 bease of If P ; bease of If the seee wll be as follows: If P ; 6 bease of If P ; 8 bease of If P ; bease of Let If the frst elemets of the seee are where for Ths we have the seee Atm Iraa Joral of See & Teholog Tras A Volme Nmber A

4 78 O Deve / E Karadma (where for 6 ) (where for 7 ) 7 7 Se the elemets seedg 6 deed o ad for ther vales the le d begs aga wth the elemet; that s Ths P ; If the the frst elemets of the seee are where for Ths we have the seee (where for ) (where for ) Se the elemets seedg deed o ad for ther vales the le d begs aga wth the elemet; that s Ths P ; If the frst elemets of the seee are where for Ths we have the seee Iraa Joral of See & Teholog Tras A Volme Nmber A Atm (where for ) Se the elemets seedg deed o ad for ther vales the le begs aga wth the d elemet; that s Ths P ; Also see [8] for a dfferet roof whe ; se Theorem Let be the gro defed b the resetato : S S The the followg are tre

5 K-a seees mller s 79 If P ; If N P ; If s a rme mber ad the P ; are the same for both ad v If ad ( ) s the bggest of rme mbers the ether P ; are the same for both ad or P ; P ; Where P ; P P ; ; ; deote erod of for ad ; meas that dvdes P P Proof: We frst ote that If the seee wll be as follows: () If P ; 6 bease of ; D If N the seee redes to a a a a Where a a N Se the elemets seedg 6 deed o for ther vales the le begs aga 6 wth the 6 elemet; that s Ths P 6 6 ; 6 If s a rme mber ad the we have the seee b 9 7 b Where b b N If 7 ad 99 or 8 ad 76 or b ad b the 7 ad 99 or 8 ad 76 or b ad b So t a be see that from () P ; are the same for both ad v B omtg b b () t a be see that ether P ; are the same for both ad or P ; P ; Let If the P ; bease of ; D If N the frst elemets of the seee are where for Ths we have the seee () Atm Iraa Joral of See & Teholog Tras A Volme Nmber A

6 8 O Deve / E Karadma 6 (where for 6 ) (where for ) (where for 9 8) 7 (where for - ad N ) (where for ad ) N Se the elemets seedg deed o for ther vales the le begs aga wth the elemet; that s Ths P ; If s rme mber ad the frst elemets of the seee are where for Ths we have the seee Iraa Joral of See & Teholog Tras A Volme Nmber A Atm

7 K-a seees mller s Atm Iraa Joral of See & Teholog Tras A Volme Nmber A 8 ) (where N () If the So t a be see that from () P ; are the same for both ad v B omtatg () t a be see that ether P ; are the same for both ad or P P ; ; The ad v aoms the Theorem are vald for both ; ad ; bease of ; ; Theorem Let be the gro defed b the resetato : S S The the followg are tre If the ; 6 P mod ; mod otherwse P Let If there s o t sh that t s a odd fator of the mod ; mod otherwse P Let be the bggest odd fator of the two ases or: If for N the mod ; mod otherwse P If s the bggest odd mber whh s ad for N the

8 8 O Deve / E Karadma mod P ; mod otherwse If s a rme mber ad P ; are the same for both ad If ad ( ) s the bggest of rme mbers the ether P ; are the same for both ad or P ; P ; the Proof: The roof s smlar to the roofs of Theorem ad Theorem REFERENCES Wall D D (96) Fboa seres modlo m Amer Math Mothl 67 - Wlo H J (986) Fboa seees of erod gros Fboa Qarterl () 6-6 Cambell C M Dooste H & Robertso E F (99) Fboa legth of geeratg ars gros : Alatos of Fboa Nmbers Vol eds A Bergm et al Klwer Dordret 7- Dooste H & Cambell C M () Fboa legths of atomorhsm gros volvg trboa mbers Vetam J Math Cambell C M Cambell P P Dooste H & Robertso E F () O the Fboa legth of owers of dhedral gros : Alato of Fboa Nmbers Vol 9 ed F T Howard Klwer Dordret D R & Smth C (99) Rerrees fte gros Trsh J Math 9() -9 7 D R & Smth C (997) Fboa seees fte lotet gros Trsh J Math () - 8 Adı H & D R (998) eeral Fboa seees fte seees fte gros Fboa Qarterl 6() 6-9 Karadma E & Yavz U () O the erod of Fboa seees lotet gros Aled Mathemats ad Comtato (-) - Karadma E & Adı H () eeral -ste Fboa seees lotet gros of eoet ad lote lass Aled Mathemats ad Comtato (-) 9-97 Karadma E & Adı H () O Fboa seees lotet gros Math Balaa 7(-) 7- Karadma E & Adı H () O the relatosh betwee the rerrees lotet ad the bomal formla Ida Joral of Pre ad Aled Mathemats (9) 9- Ko S W (99) Fboa seees fte gros Fboa Qarterl () 6- Adı H & Smth C (99) Fte -otets of some lall reseted gros J Lodo Math So 9() 8-9 Dooste H & alome R () Comtg o Fboa legths of fte gros It J Al Math () Cambell C M Cambell P P Dooste H & Robertso E F () Fboa legths for erta metal gros Algebra Collom () - 7 Dooste H & Hashem M (6) Fboa legths volvg the Wall mber ( ) J Al Math Comt (-) Karadma E & Deve Ö (9) -a seees fte tragle gros Dsrete Dams Natre ad Soet Iraa Joral of See & Teholog Tras A Volme Nmber A Atm

9 K-a seees mller s 8 9 Coeter H S M & Moser W O J (97) eerator ad relatos for dsrete gros rd edto Berl Srger Atm Iraa Joral of See & Teholog Tras A Volme Nmber A