Permutations that Decompose in Cycles of Length 2 and are Given by Monomials

Transcription

1 Poceedgs of The Natoa Cofeece O Udegaduate Reseach (NCUR) 00 The Uvesty of Noth Caoa at Asheve Asheve, Noth Caoa Ap -, 00 Pemutatos that Decompose Cyces of Legth ad ae Gve y Moomas Lous J Cuz Depatmet of Mathematcs Uvesty of Pueto Rco at Humacao Humacao, Pueto Rco Facuty Advso: D Ivesse Ruo F Astact I ths pape pemutatos of fte feds gve y moomas ax ae studed I patcua, the ecessay ad suffcet codtos the coeffcet a ad the expoet to ota pemutatos that decompose to cyces of egth ae studed Keywods: Pemutatos, Cyces, Fte Feds Itoducto A pemutato s a eodeg of the eemets a set Ou pcpa teest s to fd pemutatos of fte feds that decompose cyces of egth We study pemutatos of F, = p, p pme, that ae gve y moomas of the fom ax ad decompose cyces of egth Hee we peset some pata esuts o the ecessay ad suffcet codtos o the coeffcet a ad the expoet to ota ths type of pemutatos Pemutatos ca e used fo the costucto of teeaves fo eo coto codes Eo coto codes ae used commucatos systems to potect the fomato of eos that ca occu dug tasmsso Pemutatos that decompose cyces of egth ae patcuay usefu ecause they ae the ow vese ad hece the same techoogy ca e used fo ecodg ad decodg Pemaes We eg y pesetg the ecessay ackgoud fo the est of the pape Fte feds We ae teested pemutatos of fte feds used the est of the pape Let F e a o-empty set wth two opeatos ( ) Fo a a,, c, F oe must have, F The foowg cocepts ad esuts aout fte feds w e +, We say that F s a fed f t satsfes the foowg popetes

2 a a + ) If F ad F, the F ) a + ( + c) = ( a + ) + c ) a + = + a ) Thee s a eemet 0F F such that a + 0 = a = 0 a fo evey a F F F + a a + x = 0 F a a a ( c) = ( a ) c a = a a, a F = a = F a fo a a 0 F a x = F a ( + c) = a + a c ( a + ) c = a c + c ) Fo each F, the euato has a souto F ) If F ad F, the F ) ) fo a F ) Thee exst a eemet such that F 0 F 0) Fo each, the euato has a souto F ), ad a F A fte fed s a fed wth a fte ume of eemets It s we kow that evey fte fed has = eemets, whee p s a pme ume The oe-zeo eemets of a fte fed, F : = sge eemet caed a pmtve oot Moe fomay, F p \{0}, ca e geeated y a Defto Let F We say that s a pmtve oot of F f ad oy f geeates a the eemets of Exampe I F, s a pmtve oot Note that, Z 0, F = \{0} Ths s, F = {,,, } 0 =, =, =, =, =, =, =, =, =, =, 0 = 0, = It s kow that evey fte fed F has a pmtve oot ad t s easy to see that = The ext poposto foows easy fom ths fact Poposto Let F The, = Defto Let Z ad gcd (, ) = We say that j s the ode of Z ad wte f j s the j = od (a) j smaest postve tege such that ( mod ) Smay, we say that j s the ode of F f j s the smaest postve tege such that j (mod ) Note that, Exampe, the smaest postve tege j such that j (mod ) s Ths s ot a cocdece; fact, the ode of a pmtve oot s aways - F Pemutatos A pemutato π of a set A s a jecto π : A A Let e the fte fed wth eemets It s we kow that a mooma ax a F gves a pemutatos of F f ad oy f gcd(, -)= We ca ths type of, moomas pemutato moomas F

3 Exampe Let A = Z ad defe π : Z Z y π ( x ) = x Sce gcd(,0) =, π (x) s a pemutato Z mooma of Ths pemutato ca e epeseted the foowg way, whee a the eemets of the doma ae the fst ow ad the secod ow s the mage: Aothe way to epeset pemutatos s wth ts decomposto cyces To epeset the pemutatos ths way oe takes a ta vaue ad pace t the egg of a cyce () The take the vaue that we otaed whe evauatg π (x) ad pace t to the sde of Now takeπ () ad evauate t aga the same fucto If whe dog ths oe otas the ta vaue, the the cyce fshes ad the cyce s ( π () ) If ot, oe epeats the evauato wth the pevous esut ut the ta vaue s otaed Note that the eemets the cyce ae the esut of composg the fucto wth tsef ad evauatg t The cyce fshes whe oe otas the ta vaue Each cyce w have the fom, ( π ( ) π ( π ( )) π ( ) = ) whee π () meas π composed wth tsef tmes ad evauated, ad s the smae vaue so that π ( ) = If π ( ) =, the s caed a fxed pot ad oe does ot wte the cyce Cotug wth the pevous exampe, the cycc decomposto of the pemutato of s: ( ) ( ) Z gve y π ( x ) = x We ae teested pemutatos of F gve y moomas ax that decompose cyces of egth Fo exampe, the pemutato of gve y π ( x ) = x decomposes cyces of egth The pemutato s: Z The cycc decomposto of ths pemutato s: Pemutato Moomas ( ) ( ) ( ) ( ) (0 ) ( ) A pemutato mooma A s a mooma such that whe t s evauated the eemets of A poduces a pemutato of A Cosde F, the fte fed wth eemets It s we kow that the fucto π : F F defed y π x) = ax, a F, poduces a pemutato of F f ad oy f gcd(, -)= We ae teested ( pemutatos of F that decompose cyces of egth ad ae otaed usg moomas ax Pemutato moomas x Theoem [] gves the ecessay ad suffcet codtos to ota pemutato moomas x that decompose cyces of the same egth The foowg poposto s a cooay to Theoem ad gves the ecessay ad suffcet codtos o the expoet to ota pemutatos that decompose cyces of egth ad ae gve y moomas x

4 k k k Poposto Let = p p p, k Z, k The pemutato of gve y x decomposes cyces of the egth f ad oy f oe of the foowgs hods fo each = k p ) (mod ) ) = od ( ) k p F Exampe Cosde Z Tae ustates Poposto Tae cycc decomposto of pemutatos gve y x ad the ode of mod Cycc decomposto od ( ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( ) ( 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 ) ( ) F Aso [], Theoem gves a fomua fo coutg the ume of moomas x that poduce pemutatos of that decompose cyces of the same egth j The foowg poposto s a cooay to ths theoem ad couts the ume of moomas x that decompose cyces of egth k Poposto Let = p p, k, k Z, k 0, k The ume of pemutatos x of F that decompose cyces of egth s: k k + + f f f k = 0, k = k Note that ths poposto pedcts that thee ae moomas that poduce pemutatos of Z that decompose cyces of egth ad ths s exacty what we saw Exampe x The pevous esuts appy to case of moomas of the fom x The pupose of ths wok s to geeaze these esuts We wat to fd esuts fo moomas ax whee a F, a We foud that, some cases, ax decompose cyces of egth fo a a a F, othes, we eed addtoa codtos to ota cyces of egth fo Hee we peset some pata esuts of whe the pemutatos gve y ax decompose cyces of egth

5 Ou fst esut pesets a case whee the pemutatos gve y a F ax decompose cyces of egth fo a ( Theoem Let π x) = ax, a F The π gves a pemutato of that decomposes cyces of egth fo a a F Poof: We fst pove that ax s a pemutato mooma of F We have to see that gcd(-, -)= Suppose that d s the geatest commo dvso of - ad - The -=dk ad -=d whee k, Z Ths mpes that d=-=-+=dk+ ad d=dk+ Theefoe d(-k)= ad ths mpes that d dvdes Sce d s a postve tege, d= Hece ax s a pemutato mooma of F To costuct the cyces of the pemutato of gve y ax, a eemet F s evauated the fucto π ( x) = ax F ad the esut s evauated aga the same fucto to ota: F ( a a( a ) ) To have cyces of the egth, we must have, a( a ) = Now, a( a ) + ( )( ) + = a = a = a ( ) Usg Poposto, we have that a ( ) = ad theefoe the cyces have egth Befoe, we metoed that the moomas ax ae pemutatos moomas of F f ad oy f gcd(-,) = The ext emma gves the codto fo the moomas ax Lemma Let = p, p The ax s a pemutato mooma of f ad oy f (-) F Poof: ( ) Suppose that (-) The -=k To see that ax gves pemutato of, we must pove that gcd, = F Sce =, epacg - y k, we ota: gcd, = f ad oy f gcd(k, k-)= k = =k- Theefoe Suppose that d s the geatest commo dvso of k ad k- The, k=d ad k-=dm fo some,m Z Now mutpyg k- y, we ota k-=dm But fom the fst euato we have that k-=d- Hece, d- =dm Ths mpes that =d(-m) ad d dvdes Sce d s a postve tege, ths meas that that d= o d=

6 Sce d dvde k-, d must e Theefoe, gcd s a pemutato mooma of F, = gcd(k, k-)= Ths mpes that ax ( ) Suppose that ax s a pemutato mooma of F The gcd, = Aso p ad = p mpy that (-) Now gcd, = mpes that does ot dvde Theefoe, =k+,k Z Now sovg fo, we ota that =k++ Ths mpes that -=(k+) ad hece, (- ) It s kow that the pemutatos gve y x decompose cyces of egth (see [] ), ut whe the coeffcet of ax s ot euas to ot a the pemutatos decompose cyces of egth The ext theoem gves the ecessay ad suffcet codtos such that ax gves a pemutatos of that decompose cyces of egth F Theoem Let = p, p ad et e a pmtve oot of F The k that decompose cyces of egth f ad oy f a =, k Z ad (-) ax gves a pemutato of F Poof: ( ) k Suppose that (-) ad a = The, y the pevous emma, ax s pemutato mooma of F To costuct the cyces of the pemutato of k x π ( x) = ad the esut s evauated aga the same fucto F gve y ax, a eemet F s evauated the fucto k = k k k k To have cyces of egth, oe must have that, k k = = = = Now, k k + k ( )( ) + + k k ( ) ( ),

7 ( ) k whee = = Z ecause (-) Usg Poposto, we have that ( ) =, ad theefoe the cyces have egth ( ) Sce π ( x) = ax s pemutato of, Lemma mpes that (-) Now, et a = ad F We have two cases: ) s a cyce of egth, o ) F s a fxed pot Case ): Suppose that F s a cyce of egth The, + + = Now we smpfy + ad ota = = Now we ewte as ( ) + = ( ) =, whee = ( ) ad s a tege By Poposto we have that ( ) = + Ths mpes that = = Ths mpes that = ad hece 0 mod ( ) Theefoe k = ( ) k, k Z Theefoe, =k k Z ad a = Case ): Suppose ow that s a fxed pot Ths meas that = Hece, = = Sce - =k, k Z, j = =k-, =k-,aso, sce s a pmtve oot F, = fo some j Z j j Theefoe, ( ) ( ) h = = Ths mpes that = h, some h Z ad hece a = Cocusos ad wok pogess The esuts peseted o ths pape ae patas esuts We foud ecessaes ad suffcet codtos the coeffcet a, of some moomas ax, to ota pemutatos that decompose cyces of egth We ae st wokg o the foowg poems Gve ay expoet such that x decomposes cyces of egth, fd the ecessay ad suffcet codtos o the coeffcet a such that ax aso decompose cyces of egth Ae thee pemutatos gve y ax that decompose cyces of egth eve f the pemutato gve y does ot decompose cyces of egth? Is thee aothe expoet such that ax decompose cyces of egth fo a? Ackowedgmets a F Ths wok has ee suppoted pat y the Natoa Secuty Agecy, Gat Nume H0-0-C-0; ad the Natoa Scece Foudato CSEMS Pogam at the UPRH, Gat Nume 0 x

8 Refeeces Books Rudof Ld ad Haad Neddeete, Ecycopeda of Mathematcs ad ts Appcatos Vo 0, Fte d Feds, ed (Uted Kgdom: Camdge Uvesty Pess, ) I Ruo ad C Coada-Bavo, Cycc Decomposto of Pemutatos of Fte Feds Otaed usg Moomas ad Appcatos to Tuo Codes, Fte Feds ad Appcatos, LNCS, ed Mue, Po, Stchteoth (New Yok: Spge, 00), d Thomas Hugefod, Astac Agea: A Itoducto, ed (Foda: Saudes, ) Jouas Yaa B Lus ad Lus O Péez, Pemutatos of, costucted usg sevea mooma odegs, Poceedgs of the 00 NCUR (Ap, 00) Z p