Jurnal Teknologi NUMBER OF COMPATIBLE PAIR OF ACTIONS FOR FINITE CYCLIC GROUPS OF P-POWER ORDER. Full Paper

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1 Junal Tenologi NUMBER OF OMPATIBLE PAIR OF ATIONS FOR FINITE YLI GROUPS OF P-POWER ORDER Mohammed Khalid Shahoodh, Mohd Sham Mohamad *, Yuhani Yusof, Sahimel Azwal Sulaiman Alied & Industial Mathematics (AIMs) Reseach luste, Faculty of Industial Sciences & Technology, Univesiti Malaysia Pahang, Lebuhaya Tun Raza, 2600 Gambang, Kuantan, Pahang Daul Mamu, Malaysia Full Pae Aticle histoy Received 17 July 2017 Received in evised fom 27 Ail 2018 Acceted 15 May 2018 Published online 1 August 2018 *oesonding autho mohdsham@umedumy Gahical abstact i1 ( 1) ( 1) ( 1) ( 1) i1 Abstact The comatible actions layed an imotant ole befoe detemining the nonabelian tenso oduct of gous Diffeent comatible ai of actions gives a diffeent nonabelian tenso oduct even fo the same gou The aim of this ae is to detemine the exact numbe of the comatible ai of actions fo the finite cyclic gous of -owe ode whee is an odd ime By using the necessay sufficient numbe theoetical conditions fo a ai of the actions to be comatible with the actions that have -owe ode, the exact numbe of the comatible ai of actions fo the finite cyclic gous of -owe ode has been detemined given as a main esult in this ae Keywods: Nonabelian tenso oduct, cyclic gous, automohism gou, comatible actions, numbe theoy Absta Tindaan yang seasi memainan eanan enting sebelum menentuan umulan hasil daab tenso ta abelan Pasangan tindaan seasi yang bebeza membeian umulan hasil daab tenso yang belainan walauun daiada umulan yang sama Objetif ajian ini adalah untu menentuan jumlah sebena asangan tindaan seasi bagi umulan itaan tehingga beeingat uasa dengan ialah nombo ganjil edana Dengan menggunaan syaat elu dan cuu teoi nombo untu seasang tindaan menjadi seasi dengan tindaan yang beeingat uasa, jumlah sebena asangan tindaan seasi umulan itaan tehingga beeingat uasa telah ditentuan dan dibeian sebagai hasil utama dalam maalah ini Kata unci: Hasil daab tenso ta abelan, umulan itaan, umulan automofisma, tindaan seasi, teoi nombo 2018 Penebit UTM Pess All ights eseved 80:5 (2018) wwwjunaltenologiutmmy eissn

2 164 Shahoodh, Mohamad, Yusof & Sulaiman / Junal Tenologi (Sciences & Engineeing) 80:5 (2018) INTRODUTION The nonabelian tenso oduct of gous has been discussed since 1984 Bown Loday [1] wee the fist who intoduced the concet of the nonabelian tenso oduct of gous with the comatible actions This concet was oiginated in a connection with a genealized Van Kamen Theoem its constuction has its oigins in the algebaic K-theoy also in the homotoy theoy Given two gous G H, then the nonabelian tenso oduct G H is the gou geneated by satisfying the following elations g g h h gg h ( g h)( gh ) ghh ( gh)( g h ) g, g G h, h H g h fo all If G H be gous which act on each othe each of which acts on itself by conjugation satisfying ( g h) g g g, g G 1 g h g ( ( )) h, h H, ( h g) h h 1 h g h ( ( )) fo all then the actions ae said to be comatible A ae by Bown et al [2] motivated many eseaches to investigate seveal tyes of gous with the nonabelian tenso oduct the nonabelian tenso squae Some nown esults with an oveview on the nonabelian tenso oduct liteatue was illustated by Kae [] Many eseaches had studied the nonabelian tenso squae G G, since the actions ae tivial easily to chec the comatibility conditions If G H, then it s in needed to chec the comatible of the actions when the actions ae nontivial Ellis McDemott [4] investigated the nonabelian tenso oduct fo diffeent gous checed the comatible conditions of nontivial actions, but only focused on the quatenion gous of ode u to 2 Vissche [5] continued the study on the nonabelian tenso oduct of -owe ode, whee is ime, he focused on the cyclic gous He comuted some of the nonabelian tenso oduct of the cyclic gous of -owe ode gives a comlete classification of the nonabelian tenso oduct of cyclic gous of 2-owe ode with mutual nontivial actions of ode two Vissche [5] also ovided necessay sufficient conditions fo a ai of finite cyclic gous of -owe ode to act comatibility on each othe Next, Mohamad [6] continued with Vissche s wo he gave new necessay sufficient conditions fo a ai of finite cyclic gous of -owe ode to act comatibility on each othe with the conditions ae deend on the ode of the action Mohamad et al [7] also ovided the comatibility conditions the nonabelian tenso oduct of the 2 cyclic gous of ode with the actions of ode Thee yeas late, Sulaiman et al [8] comuted the comatible ai of nontivial actions fo some cyclic gous of 2-owe ode involving at least one of the actions having ode geate than two Sulaiman et al [9] have studied the comatible ai of nontivial actions fo two same finite cyclic 2- gous of 2-owe ode ovided the exact numbe of the comatible ai of nontivial actions fo such tye gous with the actions that have a same ode while Mohamad [10] give the numbe of comatible ai of actions fo two same cyclic gous of 2-owe ode fo all ode of actions Futhemoe, Sulaiman et al [11] give moe geneal esults which detemined the maximum numbe of diffeent nonabelian tenso oduct fo the two finite cyclic gous of 2-owe ode by comuting the exact numbe of the comatible ais of actions fo the given nonabelian tenso oduct fo such tye of gous Besides that, Shahoodh et al [12] detemined the comatible ai of nontivial actions fo cyclic gous of -owe ode Howeve, they gave some necessay sufficient numbe of theoetical conditions fo a ai of finite cyclic gous of -owe ode with nontivial actions to act comatibly on each othe with the actions that have ode thee nine satisfy the comatibility conditions In this ae, the aim is to ovide the exact numbe of the comatible ais of actions fo the finite cyclic gous of -owe ode whee is an odd ime GAP softwae [1] is used to comute the comatible actions fo the finite cyclic gous of -owe ode whee is an odd ime In section 2, some definitions elated esults on the automohism of cyclic gous of -owe ode, numbe theoy comatible actions ae given Then, the numbe of the comatible ai of actions fo the finite cyclic gous of -owe ode whee is an odd ime is given in Section Lastly, the conclusions of this ae ae given in Section 4 20 METHODOLOGY In this section, some definitions elated esults on the automohism of cyclic gous of -owe ode, numbe theoy comatible actions ae stated will be used to ove the new esults Stated with definitions of an action comatible actions that ae given as follows: Definition 21 [5] Let G H be cyclic gous An action of G on H is a maing, : GAut( H ) such that ( gg)( h) ( g)( ( g)( h )) fo all g, g G h H Definition 22 [5] Let G H be gous which act on each othe each of which acts on itself by conjugation Then the actions ae comatible if 1 ( g h) g h g ( ) g ( ( g )) h g fo all g, g G h, h H h h 1 h g h ( ( ))

3 165 Shahoodh, Mohamad, Yusof & Sulaiman / Junal Tenologi (Sciences & Engineeing) 80:5 (2018) Next, the automohism of the finite cyclic gou of -owe ode is an isomohic with a diect oduct of two finite cyclic gous as stated in following theoem Theoem 21 [14] Let be an odd ime If G is a cyclic gou of ode, then Aut( ) ( 1) 1 Aut( ) ( ) ( 1) Now, the Eule Phi-function fo a given ositive intege is stated in the following definition Definition 2 [15] Eule s φ-function Fo is the numbe of the ositive integes not exceeding m that ae elatively ime with m m 1, the Eule s Phi-function, denoted by ( ), The following coollay gives that the tivial action is always comatible with any othe action oollay 21[5] Let G H be gous Futhemoe, let G act tivially on H If G is abelian, then fo any action of H on G the mutual actions ae comatible Fo the case of two finite cyclic gous of -owe ode whee is an odd ime, the theoem below stated the comatibility fo the ai of the actions that have -owe ode Theoem 22 [6] Let whee m G g H h, 2 Futhemoe, let Aut( G ) with whee whee 1,2,, 1 Aut( H ) with 1,2,, 1 Then (, ) is a comatible ai if only if min, In the next section, all the esults fo the numbe of the comatible ai of actions fo the finite cyclic gous of -owe ode whee is an odd ime ae given 0 RESULTS AND DISUSSION In this section, the numbe of the comatible ai of actions is detemined Accoding to the ode of the actions between two finite cyclic gous of -owe ode that satisfying the comatibility conditions, the numbe of the comatible ai of actions that have -owe ode fo such tye gous has been detemined Since the finite cyclic gous of -owe ode ae consideed, then an action of a gou G on a gou H is the homomohism fom G to Aut(H) an action of a gou H on a gou G is the homomohism fom H to Aut(G) Hence, the numbe of the automohisms of such tye gous needs to be find fist Thus, the numbe of the automohisms fo the finite cyclic gous of -owe ode with esective ode is given in the following oosition Poosition 1 Let G x be a finite cyclic gou of -owe ode with is an odd ime ode Then, thee exist Poof Let whee ( 1) 1,2,, 1 1 automohism of G x be a finite cyclic gou of - owe ode with is an odd ime Without loss of geneality, suose that H be a finite cyclic -subgou of G such that whee H 1,2,, 1 Thus, each element that elatively ime with has an ode Since H be a cyclic subgou, by Definition 2, we have 1 ( ) ( 1) which give the numbe of the automohisms that have ode Next, the numbe of the comatible ai of actions fo the finite cyclic gous of -owe ode is detemined Suose that one of the actions has an ode one, then the numbe of the comatible ai of actions fo such tye of gous given in the following oosition Poosition 2 Let G H be finite cyclic gous of -owe ode Futhemoe, let Aut( G) with, 1 Then, the numbe 1 1 of the comatible ai of actions is ( 1) Poof Let G H be finite cyclic gous of -owe ode Futhemoe, let Aut( G ) with 1, 1 By oollay 21, when G act tivially on H, then any action of H on G the mutual actions ae comatible Thus, by Theoem 21, 1 Aut( H) ( 1) is the numbe of the comatible ais of actions Now, the numbe of the comatible ai of actions fo the two finite cyclic gous of -owe ode whee is an odd ime has been detemined when one of the actions has an ode whee 1,2,, 1 By using the necessay sufficient numbe theoetical conditions fo such tye gous, the numbe of the comatible ai of actions fo the secific value of is given as follows Poosition Let G H be finite cyclic gous of -owe ode Futhemoe, let Aut( G ) with whee 1,2,, 1 Aut( H ) such that, Then the numbe

4 166 Shahoodh, Mohamad, Yusof & Sulaiman / Junal Tenologi (Sciences & Engineeing) 80:5 (2018) of the comatible ais of actions is 1 1 i1 ( 1) ( 1) ( 1) when 1,2,, 1 min, i1 Poof Let G H of -owe ode Futhemoe, let that whee, be finite cyclic gous Aut( G) 1,2,, 1 Aut( H ) such By Poosition 1, thee ae ( 1) automohisms of ode whee 1,2,, 1 by Theoem 22, the actions ae comatible with Thus, we shall conside into two 1 cases as follows: ase I: Suose that 1 1, then by oollay 21, when one of the actions is tivial, then the actions ae comatible Thus, thee ae ais of actions unde this case ase II: Suose that ( 1) whee 1 comatible 1,2,, 1 by Theoem 22, the actions ae comatible min, Hence, the numbe of the when comatible ai of actions fo evey is the summation of the ossibilities of the actions to be comatible, which ae ( 1) ( 1) ( 1) ( 1) ( 1) ( 1) min, 1 ( 1) i1 ( 1) whee ( 1) i1 min, By Poosition 1, thee ae 1 automohisms of ode 1 i 1 i1 Thus, thee ae ( 1) ( 1) comatible ais of actions unde this case Theefoe, in total thee ae 1 1 i1 ( 1) ( 1) ( 1) comatible ais of i1 actions with min, The following oosition gives the total numbe of the comatible ais of actions fo two finite cyclic gous of -owe ode when one of the actions has the -owe ode Poosition 4 Let G x H y whee, Futhemoe, let ae elements of Aut( G ) Aut( H ) esectively with whee 1,2,, 1 Then the total numbe of the comatible ais of actions is i1 ( 1) ( 1) ( 1) when ,2,, 1 min, Poof Let, G x H y whee Futhemoe, suose that ae elements of Aut( G ) Aut( H ) esectively with ae whee 1,2,, 1 By Poosition, thee 1 1 i1 ( 1) ( 1) ( 1) comatible ais of actions when 1,2,, 1 i1 min, Then in total if all s ae conside then the numbe of actions ae given in the following ( 1) ( 1) ( 1) i i i1 ( 1) ( 1) ( 1) i1 ( 1) ( 1) ( 1) i1 ( 1) 1 ( 1) 1 i1 1 ( 1) ( 1) ( 1) 1 i1 1 1 i i1 ( 1) ( 1) ( 1) Theefoe, thee ae i i1 ( 1) ( 1) ( 1) i1 comatible ais of actions In geneal, the numbe of the comatible ai of actions fo the two finite cyclic gous of -owe ode whee is an odd ime fo the given nonabelian tenso oduct is given in the following theoem Theoem 1 Let G x H y be finite cyclic gous of -owe ode with, Then, thee exist i1 ( 1) ( 1) ( 1) ( 1) i1 comatible ais of actions when 1,2,, 1 min, Poof Let G x H y be finite cyclic gous of -owe ode Futhemoe, let Aut( G ) Aut( H ) with, The numbe of the comatible ais of actions with secific ode can be detemined by seaating into two cases

5 167 Shahoodh, Mohamad, Yusof & Sulaiman / Junal Tenologi (Sciences & Engineeing) 80:5 (2018) which ae follows 1 ase I: Suose that 1 whee 1,2,, 1 as By Poosition 2, when one of the actions is tivial, then the numbe of the comatible ai of actions is ase II: Suose that 1 ( 1) whee 1,2,, 1 By Poosition 4, the numbe of the comatible ai of actions ae i1 ( 1) ( 1) ( 1) i1 when 1,2,, 1 min, Hence, in total the numbe of the comatible ais of actions fo the finite cyclic gous of -owe ode whee is an odd ime ae i1 ( 1) ( 1) ( 1) ( 1) i1 By using Theoem 1, the numbe of the comatible ai of actions fo the finite cyclic gous of -owe ode whee is an odd ime has been detemined Next, an examle is given to illustate the numbe of the comatible ai of actions fo a given two finite cyclic gous of - owe ode Examle 1 Let G H 4 be finite cyclic gous of -owe ode Now, conside the actions g l of G H act on each othe such that h h h g g fo g G l, By Theoem 1 the numbe of the comatible ai of actions is hh with i1 ( 1) ( 1) ( 1) ( 1) i1 Table 1 illustated the numbe of the comatible ais of actions fo given by using GAP 4 softwae Fom Table 1, thee ae 09 comatible ais of actions fo Hence, the esult fom 4 Theoem 1 is equivalent with the numbe of the comatible ais of actions given in Table 1 Next, the numbe of the comatible ai of actions fo the finite cyclic gous of -owe ode has been esented when the gous ae the same Accoding to Theoem 1, the numbe of the comatible ais of actions is given in oollay 1 Table 1 omatible ai of actions fo actions g l h h h h 4 whee l l g g h, with the 4 fo g g oollay 1 Let G H be finite cyclic gou of -owe ode with is an odd ime Then, thee ae i1 ( 1) ( 1) ( 1) ( 1) i1 comatible ais of actions l

6 168 Shahoodh, Mohamad, Yusof & Sulaiman / Junal Tenologi (Sciences & Engineeing) 80:5 (2018) Poof Let G H be finite cyclic gou of -owe ode with is an odd ime Since G H, Theoem 1, thee ae then min, Theefoe by i1 ( 1) ( 1) ( 1) ( 1) i1 comatible ais of actions In aticulaly, the numbe of the comatible ai of nontivial actions that have -owe ode fo nonabelian tenso oduct ae equal This esult is given in the following theoem Theoem 2 Let G x H y be finite cyclic gous of -owe ode with Then the numbe of the comatible ai of nontivial actions that have -owe ode fo the nonabelian tenso oduct ae equal Poof Let, G x H y be finite cyclic gous of -owe ode with, By Poosition 4, thee ae i1 ( 1) ( 1) 1 1 i1 comatible ais of nontivial actions when 1,2,, 1 min, Since min, min,, then fo any given nonabelian tenso oduct o, the numbe of comatible ais of nontivial actions ae same 40 ONLUSION The exact numbe of the comatible ai of actions between a ai of the finite cyclic 2-gous of - owe ode whee is an odd ime has been given By using the necessay sufficient conditions fo the ai of the actions to be comatible with the actions that have -owe ode satisfying the comatibility conditions, the numbe of the comatible ai of actions is detemined This ae also gives the numbe of the comatible ai of actions fo the same such tye gous with the actions that have -owe ode The obtained esults gives that the numbe of the comatible ai of nontivial actions fo the given nonabelian tenso oduct fo such tye gous ae equal Acnowledgement This eseach is fully suoted by UMP Reseach Gant, RDU The authos fully acnowledged Univesiti Malaysia Pahang fo the aoved fund which maes this imotant eseach viable effective Refeences [1] Bown, R Loday, J L 1984 Excision homotoique en basse dimension, R Acad Sci Pais SI Math 298(15): 5-56 [2] Bown, R, Johnson, D L Robetson, E F 1987 Some omutations of Non-Abelian Tenso Poducts of Gous Jounal of Algeba 11(1): [] Kae, L 1997 Nonabelian Tenso Poducts of Gous: The ommutato onnection London Mathematical Society Lectue Note Seies [4] Ellis, G McDemott, A 1998 Tenso Poducts of Pime Powe Gous Jounal of Pue Alied Algeba 12(2): [5] Vissche, M P 1998 On the Nonabelian Tenso Poduct of Gous Dissetation State Univesity of New Yo [6] Mohamad, M S 2012 omatibility onditions Nonabelian Tenso Poducts of Finite yclic Gous of - Powe Ode PhD Thesis Univesiti Tenologi Malaysia [7] Mohamad, M S, Samin, N H, Ali, N M M, Kae, L 2012 The omutation of the Nonabelian Tenso Poduct of yclic Gou of Ode 2 Jounal Tenologi 57(1) [8] Sulaiman, S A, Mohamad, M S, Yusof, Y, Samin, N H, Ali, M M N, Ken, T L, Ahmad, T 2015 omatible Pai of Nontivial Actions fo Some yclic Gous of 2- Powe Ode The 2 nd ISM Intenational Statistical onfeence 2014 (ISM-II): Emoweing the Alications of Statistical Mathematical Sciences AIP Publishing 164: [9] Sulaiman, S A, Mohamad, M S, Yusof, Y Shahoodh, M K 2016 omatible Pai of Nontivial Action fo Finite yclic 2-Gous The National onfeence fo Postgaduate Reseach 2016, Univesiti Malaysia Pahang 005: 9-42 [10] Mohamad, M S, Sulaiman, S A, Yusof, Y Shahoodh, M K 2017 omatible Pai of Actions fo Two Same yclic Gous of 2-Powe Ode IOP onf Seies: Jounal of Physics: onf Seies 890 (2017) [11] Sulaiman, S A, Mohamad, M S, Yusof, Y Shahoodh, K M 2017 The Numbe of omatible Pai of Actions fo yclic Gous of 2-Powe Ode Intenational Jounal of Simulation, System, Sciences Technology 18(4): [12] Shahoodh, K M, Mohamad, M S, Yusof, Y Sulaiman, S A 2017 Numbe of omatible Pai of Actions fo Finite yclic Gous of -Powe Ode Intenational Jounal of Simulation, System, Sciences Technology 18(4): [1] htt://wwwga-systemogthe GAP Gou, GAP-Gous, Algoithm ogamming Vesion 47, 2015 [14] Dummit, D S Foote, R M 2004 Abstact Algeba USA: John Wiley Sons [15] Buton, D 2005 Elementay Numbe Theoy Sixth Edition USA: McGaw Hill