Commutativity in Permutation Groups

Transcription

1 Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are developed Disjoit permutatios ad disjoit collectios of permutatios are the defied i terms of trasiet poits Several commutativity results for disjoit permutatios are established Itroductio Most relevat texts defie the otios of permutatios, cycles, ad disjoit cycles o a oempty set A commutativity result regardig permutatios is the preseted However, the degree of geerality of this result varies substatially For example, some texts defie permutatios oly o fiite sets [, p 92, Defiitio 25] Furthermore, the term disjoit is defied oly for pairs of cycles, but ot for permutatios i geeral [, p 95] Thus the correspodig commutativity result is restricted to the observatio that disjoit pairs of cycles i S commute, where is a positive iteger ad S is the group of permutatios o elemets [, p 95] Other texts exted the defiitio of permutatios to a arbitrary oempty set S ([2, p 38],[3, p 77],[4, p 8]), but defie cycles oly for S o fiite sets, ad ot for Sym(S) i geeral ([2, p 40],[3, p 80],[4, p 3]) As before, the term disjoit is defied oly for pairs of cycles ([2, p 4],[3, p 8],[4, p 3]) Cosequetly, the commutativity result obtaied is restricted as above to the statemet that disjoit pairs of cycles i S commute ([2, p 4],[3, p 82, Theorem 62],[4, p 3, Lemma 32]) Still other texts also use the more geeral defiitio of a permutatio o a arbitrary oempty set [5, p 26] while restrictig the defiitio of cycles to fiite sets [5, p 46, Defiitio 6] However, the term disjoit is applied to geeral permutatios rather tha beig limited to cycles, ad is eve exteded to fiite collectios of permutatios, but is limited to permutatios i S o a fiite set [5, p 47, Defiitio 62] Furthermore, the related commutativity result is still restricted to pairs of permutatios i fiite collectios oly, ad is ot exteded to iclude arbitrary collectios or eve fiite collectios of permutatios as a whole Thus the commutativity result stated is that disjoit pairs of permutatios i S commute [5, p 47] Fially, some texts defie both permutatios [6, p 30] ad cycles [6, p 79] o arbitrary oempty sets The term disjoit is defied for arbitrary collectios of cycles [6, p 79], but ot for arbitrary collectios or eve pairs of geeral permutatios Oce agai, however, the correspodig commutativity Joural of Mathematical Scieces & Mathematics Educatio, Vol 6 No 2

2 result refers oly to pairs of cycles, statig that disjoit pairs of cycles i Sym(S) commute [6, p 79, o 0] Each of these sources restricts the term disjoit to either pairs of permutatios, cycles oly, permutatios o a fiite set, or some combiatio of these Cosequetly the commutativity result produced is limited i oe way or aother i each case This paper geeralizes these results i all three aspects by extedig the term disjoit to apply to arbitrary collectios of geeral permutatios o ay oempty set The correspodig result o commutativity is the developed i this more geeral framework Throughout this paper it is assumed that S is a oempty set Prelimiary Results We begi with some basic defiitios pertiet to all of the followig results The iitial defiitios of permutatios, Sym(S), S, cycles, ad the idetity map o S are stadard, ad are icluded here for completeess Defiitio : If S is a oempty set, the a permutatio (or symmetry) o S is a -, oto fuctio :SS The set of all permutatios o S is deoted by Sym(S) If S is a fiite set of order the Sym(S) will be writte S, ad is called the set of permutatios o elemets I this case S ca be represeted as S = k k If is a positive iteger, the a permutatio Sym(S) is a cycle of legth if ad oly if there is a fiite subset i i a of S such that ( a i ) a i for i, ( a ) = a, ad (x) = x for each xs a i i I this case we write = a,a 2,, a The idetity map o S is deoted by S It is commoly kow that Sym(S) edowed with the operatio of compositio of fuctios is a group [2, p 38, Theorem 6], called the group of permutatios o S It is also well kow that Sym(S) is oabelia whe S 3 ([, p 94, Theorem 220],[2, p 40, Theorem 63]) Therefore ay otrivial result o commutativity i permutatio groups is sigificat I order to achieve the goal of this paper, we eed the stadard cocept of fixed poits, alog with a cotrastig otio of trasiet poits Thus we have the followig defiitios Defiitio 2: Suppose S is a oempty set, p,qs, ad Sym(S) The p is a fixed poit of if ad oly if (p) = p; q is a trasiet poit of if ad oly if (q) q The set of fixed poits of is F = x S (x) x I cotrast, the set of trasiet poits of is T = x S (x) x For each Sym(S), it is clear that S F = T ad S T = F Cosequetly, if F ad T are oempty, the F, T partitios S We formally state these facts i the followig result Joural of Mathematical Scieces & Mathematics Educatio, Vol 6 No 2 2

3 Corollary 3: Suppose Sym(S) (a) F ad T are set complemets of each other relative to S (b) If F ad T are oempty, the F, T is a partitio of S (a) It is clear from Defiitio 2 that F S ad T S Furthermore, for each xs, x F if ad oly if (x) = x if ad oly if x T The result follows (b) The result is a immediate cosequece of part (a) Part (a) of Theorem 4 could be stated bicoditioally However, a stroger versio of the coverse of part (a) exists, ad is stated separately i part (b) I this maer, the hypothesis i part (b) assumes oly that F for some iteger, rather tha for each iteger Cosequetly the result of the coverse of part (a) is obtaied usig a weaker hypothesis Parts (a) ad (b) are the combied to verify the result i part (c) Theorem 4: Suppose Sym(S) ad xs (a) If x F, the F for each iteger (b) Coversely, if F for some iteger, the x F (c) If F for some iteger, the F for each iteger (a) If x F, the = x by Defiitio 2 Thus for each iteger, ad [ ] = (x) = [(x)] = Hece Defiitio 2 (b) If Defiitio 2 Furthermore F by (x)s F for some iteger, the [ ] = accordig to = = [ (x) ] = ([ x Hece x F by Defiitio 2 (c) If the Sym(S) Therefore (x) = (x) [ ]) = (x)] = = S (x) = F for some iteger, the x F by part (b) Sice x F F for each iteger by part (a) Trasiet poits have properties aalogous to those i Theorem 4 for fixed poits I order to establish these properties, we apply Corollary 3 ad Theorem 4 Joural of Mathematical Scieces & Mathematics Educatio, Vol 6 No 2 3

4 Corollary 5: Suppose Sym(S) ad xs (a) If x T, the T for each iteger (b) Coversely, if T for some iteger, the x T (c) If T for some iteger, the T for each iteger (a) If x T, the x F by Corollary 3 Therefore by (the cotrapositive of) Theorem 4(b) Hece by Corollary 3 (b) If T for some iteger, the F for each iteger T for each iteger F by Corollary 3 Thus x F by (the cotrapositive of) Theorem 4(a) Hece x T by Corollary 3 (c) If the T for some iteger, the x T by part (b) Sice x T T for each iteger by part (a) Aalogous special cases of Theorem 4 ad Corollary 5 will be useful More specifically, parts (a) ad (b) of Theorem 4 ad Corollary 5 are each codesed to a sigle bicoditioal statemet for the case i which = Thus we have the followig corollary Corollary 6: Suppose Sym(S) ad xs (a) The x F if ad oly if (x) F (b) Furthermore, x T if ad oly if (x) T (a) If x F, the F by Theorem 4(a) with = Coversely, if (x) F, the x F by Theorem 4(b) with = (b) If x T, the T by Corollary 5(a) with = Coversely, if (x) T, the x T by Corollary 5(b) with = Alteratively, x T if ad oly if x F (by Corollary 3) if ad oly if (x) F (by part (a)) if ad oly if (x) T (by Corollary 3) We ow preset the defiitios of disjoit permutatios, disjoit cycles, ad disjoit collectios of permutatios i Defiitio 7 These cocepts are defied i terms of trasiet poits, ad are crucial to obtaiig the mai results o commutativity Joural of Mathematical Scieces & Mathematics Educatio, Vol 6 No 2 4

5 Defiitio 7: Suppose,Sym(S) The ad are disjoit if ad oly if T T = I particular, if = a,, a k ad = b,, b m are cycles i Sym(S), the ad are disjoit if ad oly if ai b j for each i ad j such that i k ad j m A collectio C of permutatios i Sym(S) is disjoit if ad oly if ad are disjoit for each,c such that The last part of Defiitio 7 raises the questio of whether or ot a permutatio ca be disjoit with itself That is, if S is a oempty set ad Sym(S), are ad disjoit? We resolve this issue with the followig corollary Corollary 8: Suppose Sym(S) The is disjoit with itself (that is, ad are disjoit) if ad oly if = S Sice S (x) = x for each xs the F = S by Defiitio 2, so T S = by S Corollary 3 Therefore T S T S =, so that S is disjoit with itself accordig to Defiitio 7 However, if S the there exists xs such that (x) x Thus x T by Defiitio 2, so that T Cosequetly T T = T, ad so is ot disjoit with itself Mai Results The most commo commutativity result for permutatios foud i literature states that disjoit pairs of cycles i S commute ([2, p 4],[3, p 82, Theorem 62],[4, p 3, Lemma 32]) The followig theorem geeralizes this statemet i two ways The result for disjoit cycles is exteded to disjoit permutatios i geeral Furthermore, the restrictio to permutatios i S o a fiite set cotaiig elemets is geeralized to permutatios i Sym(S) o a arbitrary oempty set S Theorem 9: Suppose,Sym(S) If ad are disjoit, the = If ad are disjoit permutatios i Sym(S), the T T = by Defiitio 7 Furthermore, for each xs, either x T, x T, or xs( T T ) However, S( T T ) = (S T )(S T ) = F F by Corollary 3 If x T the T by Corollary 5(a) (or Corollary 6(b)) The x,(x) T sice T T = Therefore x,(x) F by Corollary 3, so that (x) = x ad [(x)] = (x) by Defiitio 2 Thus (x) = [(x)] = (x) = [(x)] = (x) Similarly, if x T the = (x) Fially, if x F F, Joural of Mathematical Scieces & Mathematics Educatio, Vol 6 No 2 5

6 the = x ad (x) = x by Defiitio 2 Therefore (x) = [(x)] = (x) = x = (x) = [(x)] = (x) Cosequetly (x) = (x) for each xs Hece = I the proof of Theorem 9, DeMorga s Laws ad Corollary 3 were used to show that S( T T ) = F F A similar applicatio of these two results provides a slightly differet perspective o the result of Theorem 9 More specifically, the same commutativity result established i Theorem 9 ca be obtaied by a relatioship betwee the sets of fixed poits of permutatios rather tha their respective sets of trasiet poits Corollary 0: Suppose,Sym(S) If F F = S, the = Note that F F = S if ad oly if S( F F ) = if ad oly if (S F )(S F ) = if ad oly if T T = (by Corollary 3) if ad oly if ad are disjoit (by Defiitio 7) Thus if F F = S, the ad are disjoit Hece = by Theorem 9 Theorem 9 exteded the commo commutativity result for pairs of cycles i S o a fiite set to the same result for pairs of geeral permutatios i Sym(S) o a arbitrary oempty set We ow geeralize the third aspect of this result for disjoit pairs of permutatios i Sym(S) by extedig it to iclude disjoit collectios of permutatios i Sym(S) Corollary : If C is a disjoit collectio i Sym(S), the = for each,c Suppose C is a disjoit collectio i Sym(S) ad,c Therefore either = or ad are disjoit by Defiitio 7 If =, the clearly = Otherwise ad are disjoit, ad so = by Theorem 9 Cocludig Remarks It should be oted that the coverse of Theorem 9 is ot true That is, if,sym(s) ad =, the it is ot ecessarily true that ad are disjoit For a simple example, suppose that S ad S Clearly (like ay permutatio) commutes with itself However, is ot disjoit with itself by Corollary 8 Furthermore, the coverse of Theorem 9 is false eve i the case of distict permutatios o a fiite set For example, suppose S = {,2,3,4}, Joural of Mathematical Scieces & Mathematics Educatio, Vol 6 No 2 6

7 2 3 4, S 4, = 2 3 4, ad = Therefore = = However, T = {,2} ad T = {,2,3,4} Therefore T T = {,2}, so that ad are ot disjoit by Defiitio 7 Couterexamples also exist with permutatios o ifiite sets Suppose R is the set of real umbers, is a odd iteger, ad 3 Defie (x) = x ad (x) = x for each xr Therefore,Sym(R) ad (x) = x = (x) for each xr, so that = However, F = F = {0,,}, so that T = T = R{0,,} by Corollary 3 Hece T T = R{0,,}, ad so ad are ot disjoit accordig to Defiitio 7 Richard Wito, PhD, Tarleto State Uiversity, Texas, USA Refereces Burto, David M, Abstract Algebra, Wm C Brow Publishers, Dubuqua, Iowa, Durbi, Joh R, Moder Algebra: A Itroductio, 3rd editio, Joh Wiley & Sos, New York, Gallia, Joseph A, Cotemporary Abstract Algebra, D C Heath ad Compay, Lexigto, Massachusetts, Herstei, I N, Abstract Algebra, Macmilla Publishig Compay, New York, Hugerford, Thomas W, Algebra, Spriger-Verlag, New York, Shapiro, Louis, Itroductio to Abstract Algebra, McGraw-Hill Book Compay, New York, 975 Joural of Mathematical Scieces & Mathematics Educatio, Vol 6 No 2 7