Jurnal Teknologi THE SQUARED COMMUTATIVITY DEGREE OF DIHEDRAL GROUPS. Full Paper

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1 Jural Tekologi THE SQUARE COMMUTATIVITY EREE OF IHERAL ROUPS Muhaizah Abdul Hamid a, Nor Muhaiiah Mohd Ali a*, Nor Haiza Sarmi a, Ahmad Erfaia b, Fadila Normahia Abd Maaf a a epartmet of Mathematical Scieces, Faculty of Sciece, Uiversiti Tekologi Malaysia, 0 UTM Johor Bahru, Johor, Malaysia b epartmet of Pure Mathematics, Faculty of Mathematical Scieces ad Ceter of Ecellece i Aalysis o Algebraic Structures, Ferdowsi Uiversity of Mashhad, Mashhad, Ira Full Paper Article history Received 0 February 05 Received i revised form 5 October 05 Accepted October 05 *Correspodig author ormuhaiiah@utmmy raphical abstract THE SQUARE COMMUTATIVITY EREE OF A ROUP: {(, y) Î :( y) = ( y) P ( ) = Abstract The commutativity degree of a fiite group is the probability that a radom pair of elemets i the group commute Furthermore, the -th power commutativity degree of a group is a geeralizatio of the commutativity degree of a group which is defied as the probability that the -th power of a radom pair of elemets i the group commute I this paper, the -th power commutativity degree for some dihedral groups is computed for the case equal to, called the squared commutativity degree Keywords: Commutativity degree; ihedral group; fiite group Abstrak arjah kekalisa tukar tertib bagi suatu kumpula terhigga adalah kebaragkalia bahawa sepasag usur yag dipilih secara rawak dari kumpula tersebut adalah kalis tukar tertib Selai itu, kuasa ke- bagi darjah kekalisa tukar tertib bagi suatu kumpula adalah pegitlaka darjah kekalisa tukar tertib bagi suatu kumpula yag ditakrifka sebagai kebaragkalia bahawa kuasa ke- bagi sepasag usur yag dipilih secara rawak dari kumpula tersebut adalah kalis tukar tertib alam kajia ii, kuasa ke- bagi darjah kekalisa tukar tertib bagi kumpula dwihedro ditetuka yag maa bersamaa dega, disebut sebagai kuasa dua darjah kekalisa tukar tertib Kata kuci: arjah kekalisa tukar tertib, kumpula wihedro, kumpula terhigga 0 Peerbit UTM Press All rights reserved 0 INTROUCTION Commutativity degree is the term that is used to determie the abeliaess of groups If is a fiite group, the the commutativity degree of, deoted by P(), is the probability that two radomly chose elemets of commute The first appearace of this cocept was i 944 by Miller After a few years, the idea to compute P() for symmetric groups has bee itroduced by both Erdos ad Tura i 9 Mohd Ali ad Sarmi i 00 eteded the defiitio of commutativity degree of a group ad defied a ew geeralizatio of this degree which is called the -th commutativity degree of a group, P() where it is equal to the probability that the -th power of a radom elemet commutes with aother radom elemet from the same group They 7: (0) wwwjuraltekologiutmmy eissn 0 7

2 4 Muhaizah Abdul Hamid et al / Jural Tekologi (Scieces & Egieerig) 7: (0) determied P() for geerator -groups of ilpotecy class two A few years later, Erfaia et al 4 gave the relative case of -th commutativity degree They idetify the probability that the -th power of radom elemet of a subgroup, H commutes with aother radom elemet of a group, deoted as P ( H, ) I this research, the commutativity degree is further eteded by defiig a cocept called the probability that the -th power of a radom pair of elemets i the group commute, deoted as P () However, the focus of this research is oly for the determiatio of P (), where ad is some ihedral groups Here, P () is called the squared commutativity degree 0 PRELIMINARIES I this sectio, some importat defiitios which are the otio of commutativity degree ad its geeralizatio is stated efiitio The Commutativity egree of a roup Let be a fiite group The commutativity degree of a group, is give as: efiitio The -th Commutativity egree of a roup Let be a fiite group The -th commutativity degree of a group, is give as: efiitio 5 ihedral roups of egree For, is deoted as the set of symmetries of a regular -go Furthermore, the order of is or equivaletly = The ihedral groups, ca be represeted i a form of geerators ad relatios give i the followig represetatio: efiitio 4 -th Cetralizer of a i Let a be a fied elemet of a group The -th cetralizer of a i, C( a) is the set of all elemets i that commute with a I symbols, C a g ga a g C a Here C a C a, where C a is a subgroup of ad defie T a g ga ag ad a a Now T T a It is easy to see that T P ( ) a, y y y a b a b ba a b a may ot be a subgroup of But it ca be see easily that,, a b a b ba a b,, T C ad so T To prove all g, ag ga is a ormal subgroup of T C, let Therefore Hece a a g a g a ad so a T The for g a g a ag g a ad a C To see C T, let a C The for all g, ag ga Therefore aag ag a ad so ag a ag a Hece ag ga ad a T efiitio 5 -th Ceter of a roup The -th ceter Z ( ) of a group is the -th power of the set of elemets i that commute with every elemet of I symbols, Z ( ) { a ( a) ( a) for all i 0 RESULTS AN ISCUSSION This sectio start by defiig a ew defiitio as follow: efiitio The -th Power Commutativity egree Let be a fiite group The -th power commutativity degree of a group, is give as: Whe, the P ( ) = {(, y) Î :( y) = ( y) is called the squared commutativity degree of a group Net, the followig propositios are give which play a importat role i the proof of Theorems Propositio plays a importat role i the proof of Theorem as well as Propositio i provig Theorem Propositio {(, y) Î : ( y) = ( y) P ( ) =

3 47 Muhaizah Abdul Hamid et al / Jural Tekologi (Scieces & Egieerig) 7: (0) Let be a ihedral group of order where, ad is eve If / is odd, the T ( ) Meawhile, if / is eve, the T ( ) 4 i which Z ( ), T ( ) { g : ( g ) ( g ) ad Z a a a ( ) ( ) ( ) Case : / is odd Let where ad / is odd Suppose A e, a,, a ad,,,, B b ab a b a b Recall efiitio the we T ( e) T ( a) T ( a ) sice for all y, z A, we also yz zy ad ( yz) ( zy ) Furthermore, for all y A, ad z \ A we ( yz) ( zy) We also T ( b) { e, a, a,, a, b, a b T ( a b) T ( ab) { e, a, a,, a, ab, a b T ( a b) T ( a b) { e, a, a,, a, a b, a b T ( a b) T ( a b) { e, a, a,, a, a b, a b T ( a b) Sice for y Bad all z B, there are oly two pairs of elemets which satisfy ( yz) ( zy) e Note that Z ( ) T ( ), therefore Z ( ) { e, a, a,, a implies Z ( ) / Assume that Z ( ), therefore we T ( ) Z ( ) P Z thus P Meawhile, Q Z thus Q Therefore, P Q Note that, if ( ), if ( ), Case : / is eve Let where ad / is eve Suppose A e, a,, a ad B b, ab, a b,, a b Recall efiitio the we T ( e) T ( a) T ( a ) sice for all y, z A, we also yz zy ad ( yz) ( zy ) Furthermore, for all y A, ad z \ A we ( yz) ( zy ) We also Sice for 4 4 T ( b) { e, a, a,, a, b, a b, a b, a b 4 4 T ( a b) T ( a b) T ( a b) 4 4 T ( ab) { e, a, a,, a, ab, a b, a b, a b 4 4 T ( a b) T ( a b) T ( a b) T ( a b) { e, a, a,, a b, a b, a b, a b 4 T ( a b) T ( a b) T ( a b) y Bad all z B, there are oly four pairs of elemets that satisfy( yz) ( zy ) which cotai the elemets of e ad a / Note that Z ( ) T ( ), Z ( ) / Z ( ) { e, a, a,, a implies Assume that Z ( ), therefore we T ( ) Z ( ) 4 Eample : Let ad { e, a, a, a, a, a, b, ab, a b, a b, a b, a b The we, T ( e) { e, a, a, a, a, a, b, ab, a b, a b, a b, a b T ( a) { e, a, a, a, a, a, b, ab, a b, a b, a b, a b T ( a ) { e, a, a, a, a, a, b, ab, a b, a b, a b, a b T ( a ) { e, a, a, a, a, a, b, ab, a b, a b, a b, a b T ( a ) { e, a, a, a, a, a, b, ab, a b, a b, a b, a b T ( a ) { e, a, a, a, a, a, b, ab, a b, a b, a b, a b 4 5 T ( b) { e, a, a, a, a, a, b, a b T ( a b) T ( ab) { e, a, a, a, a, a, ab, a b T ( a b) T ( a b) { e, a, a, a, a, a, a b, a b T ( a b) 4 5 Here Z ( ) T ( ) { e, a, a, a, a, a For Z ( ), T ( ) Z ( ) implies that ( ) T Eample : 7 7 Let ad { e, a, a,, a, b, ab, a b,, a b where The we,

4 4 Muhaizah Abdul Hamid et al / Jural Tekologi (Scieces & Egieerig) 7: (0) T ( e) { e, a, a,, a, b, ab, a b,, a b 7 7 T ( a) { e, a, a,, a, b, ab, a b,, a b 7 7 T ( a ) { e, a, a,, a, b, ab, a b,, a b 7 7 T ( a ) { e, a, a,, a, b, ab, a b,, a b T ( a ) { e, a, a,, a, b, ab, a b,, a b 7 4 T ( b) { e, a, a,, a, b, a b, a b, a b 4 T ( a b) T ( a b) T ( a b) T ( ab) { e, a, a,, a, ab, a b, a b, a b 5 7 T ( a b) T ( a b) T ( a b) Here Z ( ) T ( ) { e, a, a, a, a, a, a, a For Z ( ), T ( ) Z ( ) 4 implies that ( ) T 4 Propositio Let be a ihedral group of order where ad is odd The T ( ) Let where ad / is odd Suppose A e, a,, a ad,,,, B b ab a b a b Recall efiitio the we T ( e) T ( a) T ( a ) sice for all y, z A, we also yz zy ad ( yz) ( zy ) Furthermore, for all y A, ad z \ A we ( yz) ( zy ) We also T ( b) { e, a, a,, a, b, T ( ab) { e, a, a,, a, ab, T ( a b) { e, a, a,, a, a b,, T ( a b) { e, a, a,, a, a b Sice for y Bad all z B, there is oly oe pair of elemets which satisfy ( yz) ( zy) e Note that Z ( ) T ( ), Z ( ) { e, a, a,, a implies Z ( ) / Assume that Z ( ), therefore we T ( ) Z ( ) Eample : Let ad { e, a, a, b, ab, a b The we, T ( e) { e, a, a, b, ab, a b T ( a) { e, a, a, b, ab, a b T ( a ) { e, a, a, b, ab, a b T ( b) { e, a, a, b T ( ab) { e, a, a, ab T ( a b) { e, a, a, a b Here Z ( ) T ( ) { e, a, a For Z ( ), T Z ( ) T implies that The mai results of this research are stated i the followig two theorems Theorem Let be ihedral groups of order where ad is eve 4 Z ( ) 4 i If is odd the P ( ) 4 4 Z ( ) ii If is eve the P ( ) 4 By efiitio, we P ( ) (, y) : ( y) ( y) y ( y) ( y) T () Z ( ) Z ( ) T ( ) T ( ) Z ( ) T( ) Z ( )

5 49 Muhaizah Abdul Hamid et al / Jural Tekologi (Scieces & Egieerig) 7: (0) i By Propositio (for / is odd), ii P ( ) Z ( ) Z ( ) 4 Z ( ) 4 Z ( ) 4 4 Z ( ) 4 4 By Propositio (for / is eve), P ( ) Z ( ) 4 Z ( ) 4 Z ( ) 4 Z ( ) 4 4 Z ( ) 4 Theorem Let be ihedral groups of order where If is odd the 4 Z ( ) P ( ) 4 By efiitio, we (, y) : ( y) ( y) P ( ) y ( y) ( y) T ( ) T ( ) Z ( ) Z ( ) T () Z ( ) T( ) Z ( ) By Propositio, P ( ) Z ( ) Z ( ) 4 Z ( ) 4 Z ( ) 4 4 Z ( ) 4 40 CONCLUSION The research determied the squared commutativity degree of ihedral groups The results bee foud for eve ad odd However, for eve, the result has bee divided ito two cases where / is odd ad / is eve Ackowledgemet The authors would like to ackowledge Miistry of Educatio (MOE) Malaysia ad Research Maagemet Cetre (RMC), Uiversiti Tekologi Malaysia for the fiacial fudig through the Research Uiversity rat (RU) Vote No 0H07 ad No 0H The first author would also like to thak Miistry of Educatio (MOE) for her MyPh scholarship Refereces [] Miller, 944 A Relative Number of No-Ivariat Operators i a roup, Proc Nat Acad SciUSA 5- [] Erdos, P ad Tura, P 9 O Some Problems of a Statistical roup Theory IV, Acta Math Acad Sci Hugaricae 9:4-45 [] Mohd Ali, N M ad Sarmi, N H 00 O Some Problems i roup Theory of Probabilistic Nature Meemui Matematik (iscoverig Mathematics) ():5-4 [4] Erfaia, A, Tolue, B ad Sarmi, N H 0 Some Cosideratios o the -th Commutativity egree Ars Combiatorial Joural : [5] ummit, S ad Foote, R M 04 Abstract Algebra, Third Editio Joh Wiley ad So, USA 004 [] Mashkouri, M ad Taeri B 0 O a raph Associated to roups Bulleti of The Malaysia Mathematical Scieces Society 4():55-50