Intersection Matrices Associated With Non Trivial Suborbit Corresponding To The Action Of Rank 3 Groups On The Set Of Unordered Pairs

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1 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUE 3, ISSUE, DECEBER 4 ISSN Intersecton atrces Assocated Wth Non Trva Suborbt Correspondng To The Acton Of Rank 3 Groups On The Set Of Unordered Pars BettyChepkorr, John K. Rotch, Benard C. Tonu, ReubenC. Langat Abstract: In ths paper we fnd ntersecton numbers and ntersecton matrces assocated wth each non trva sub orbt correspondng to the acton of rank 3 groups; The symmetrc group S 5,aternatng group A 5 and The dhedra group D 5 on the set of unordered pars. We showed that the coumn sum of the ntersecton matrces assocated wth s equa the ength of the suborbt. They are aso square matrces and of order 3x3. Index Terms: Intersecton atrces,non Trva Suborbt, Acton of Rank 3 Groups,Set of Unordered Pars INTRODUCTION In 964, Hgman ntroduced the rank of a group when he worked on fnte permutaton groups of rank 3. He showed that f G s a group actng transtvey on a set, where n and f G s a rank 3 group of degree n=k +, where k s the ength of a G x orbt, x then n = 5,, 5 or 35.In 97, Hgman cacuated the rank and subdegrees of the symmetrc group Sn actng on a eement subsets from the set ={,,,n}.he showed that the rank s 3 and the n,, subdegrees are n. In 97, Cameron worked on suborbts of mutpy permutaton groups and ater n 974, he studed suborbts of prmtve groups. In 978, he deat wth the orbts of permutaton groups of unordered pars. In 977, Neuman extended the work of Hgman and Sms to fnte permutaton groups, edge cooured graphs and aso matrces. INTERSECTION ATRICES ASSOCIATED WITH THE ACTION OF G S. 5 ON. Intersecton matrx correspondng to.,. Takng a ={,} n () and G {,} - orbts arranged as foows,,,,3,,4,,5,,3,,4,,5, 3,4, 3,5, 4,5 We therefore arrange,3,3 Gb - orbts as foows,3,,,4,,5, 3,, 3,4, 3,5,3,4,,5, 4,5 Betty Chepkorrs a ecturer at Unversty of Kabanga, Kenya PH E-ma: bettyrono@gma.com John K. Rotch s a ecturer at Unversty of Kabanga, Kenya and pursung PhD degree program n Apped athematcs n Unversty of Edoret, Kenya, PH E-ma: kmrott@gma.com Benard C. Tonu, s a ecturer at Unversty of Kabanga, Kenya Reuben C. Langats a ecturer at Unversty of Kabanga, Kenya 3,4 3,4 3,4 3,, 3,, 3,5, 4,, 4,, 4,5 3,4,,,5,,5 From defnton 6, the ntersecton numbers reatve to the suborbt, are defned by b,, b, Hence we fnd the ntersecton numbers reatve to, as IJSTR 4

2 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUE 3, ISSUE, DECEBER 4 ISSN foows I,, I,, 6 I,, I,3, I,3, 3 I,3, I 3,4, I 3,4, 4 I 3,4, By defnton 7 the ntersecton matrx,, where are the ntersecton assocated wth, numbers reatve to s obtaned as foows;, Intersecton matrx correspondng to, From defnton 6, the ntersecton numbers reatve to the suborbt, are defned by b b,,, We therefore fnd the ntersecton numbers reatve to, as foows I,, I,, I,, 3 I,3, I,3, I,3, I 3,4, I 3,4, I 3,4, By defnton 7 the ntersecton matrx assocated wth, numbers reatve to,, where are the ntersecton s obtaned as foows;, 3.3 Propertes of the ntersecton matrces assocated wth, and, The coumn sum of the ntersecton matrx assocated wth s equa to the ength of the suborbt.we can see that the coumn sum of s 6 aso the coumn sum of s 3. and are square matrces of order 3 IJSTR 4

3 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUE 3, ISSUE, DECEBER 4 ISSN INTERSECTION ATRICES ASSOCIATED WITH THE ACTION OF G A5 ON 3. Intersecton matrx correspondng to, we take a ={,} n () and G {,} - orbts arranged as foows,,,,3,,4,,5,,3,,4,,5, 3,4, 3,5, 4,5 We therefore arrange,3,3 Gb - orbts as foows,3,,,4,,5, 3,, 3,4, 3,5,3,4,,5, 4,5 3,4 3,4 3,4 3,, 3,, 3,5, 4,, 4,, 4,5 3,4,,,5,,5 From defnton..6., the ntersecton numbers reatve to the suborbt, are defned by b,, b, Hence we fnd the ntersecton numbers reatve to, as foows I,, I,, 6 I,, I,3, I,3, 3 I,3, I 3,4, I 3,4, 4 I 3,4, By defnton..6. the ntersecton matrx assocated wth, numbers reatve to,, where are the ntersecton s obtaned as foows;, Intersecton matrx correspondng to, From defnton..6., the ntersecton numbers reatve to the suborbt, are defned by b b,,, We therefore fnd the ntersecton numbers reatve to, as foows I,, I,, I,, 3 I,3, I,3, I,3, IJSTR 4 3

4 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUE 3, ISSUE, DECEBER 4 ISSN I 3,4, I 3,4, I 3,4, By defnton..6. the ntersecton matrx assocated wth, numbers reatve to,, where are the ntersecton s obtaned as foows;, Propertes of the ntersecton matrces assocated wth,, and,. The coumn sum of the ntersecton matrx assocated wth s equa to the ength of the suborbt.we can see that the coumn sum of s 6 aso the coumn sum of s 3. and are square matrces of order 3 4INTERSECTION ATRICES ASSOCIATED WITH THE ACTION OF G=D 5 ON By Defnton 6, gven an arrangement of the G a -orbts, the G b orbts are arranged such that f b and g(a) = b then, g a g b b 4. Intersecton matrx correspondng to. Takng a = n and G -orbts arranged as foows,.,5, 3,4, We arrange the G b - orbts as foows:.,3, 4,5, ,5, 3,4, From defnton 6, the ntersecton numbers reatve to the suborbt are defned by b, b Hence we fnd the ntersecton numbers reatve to foows I I I I I I 3 I 3 I 3 I By defnton 7 the ntersecton matrx as,, IJSTR 4 4

5 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUE 3, ISSUE, DECEBER 4 ISSN assocated wth, numbers reatve to where are the ntersecton s obtaned as foows; 4. Intersecton matrx correspondng to From defnton..6., the ntersecton numbers reatve to the suborbt are defned by b b, We therefore fnd the ntersecton numbers reatve to as foows I I I I I I 3 I 3 I 3 I By defnton 6 the ntersecton matrx assocated wth numbers reatve to,, where are the ntersecton s obtaned as foows; 4.3 Propertes of the ntersecton matrces assocated wth, and. The coumn sum of the ntersecton matrx assocated wth s equa to the ength of the suborbt.we can see that the coumn sum of s 6 aso the coumn sum of s 3. and are square matrces of order 3 5 CONCLUSION We concude thatintersecton matrces assocated wth the acton of rank 3 on aresquare matrces of order 3x 3 and that the coumn sum of the ntersecton matrces assocated wth s equa the ength of the suborbt. REFERENCES [] Akbas,.. Suborbta graphs for moduar group, Buetn of the London mathematca socety 33: [] Burnsde, W. 9. Theory of groups of fnte order, Cambrdge Unversty Press, Cambrdge (Dover reprnt 955). [3] Bon, J. V. and Cohen, A Lnear groups and dstancetranstve groups, European Journa of combnatores : [4] Cameron, P. J. 97. Permutaton groups wth mutpy transtve suborbtsi. Proc. London ath. Soc. 3 (3): [5] Cameron, P. J Permutaton groups wth mutpy transtve suborbts II. Bu. London ath. Soc. 6: [6] Cameron, P. J Orbt of permutaton groups on Unordered sets. J. London. ath. Soc. 7 (): [7] Chartrand, G Apped and agorthmc graph theory. Internatona seres n pure and apped athematcs: [8] Coxeter, H. S y graph, proceedngs of London mathematca socety 46:7-36. [9] Faradžev, I. A. and Ivanov, A. A. 99. Dstance-transtve representatons of group G wth PSL (q)<-g< PTL (,q), European ourna of combnatores : [] Harary, F Graph Theory. Addson Wesey Pubshng Company, New York. [] Hgman, D. G Fnte permutaton groups of rank 3. ath Zetschrff 86: [] Hgman, D. G. 97. Characterzaton of fames of rank 3 permutaton groups by subdegrees. I. Arch. ath : IJSTR 4 5

6 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUE 3, ISSUE, DECEBER 4 ISSN [3] Jones, G. A.; Sngerman, D. and Wcks, K., 99. Generazed Farey graphs n groups, St. Andrews 989, Eds. C. Campbe and E.F. Robertson, London mathematca socety ecture notes seres 6, CambrdgeUnversty, Cambrdge [4] Kamut, I. N. 99. Combnatora formuas, nvarants and structures assocated wth prmtve permutaton representatons of PSL (, q) and PGL (, q). Ph. D. Thess, Southampton Unversty, U.K. [5] Kamut, I. N. 6. Subdegrees of prmtve permutaton representaton of PGL (,q), East Afrcan ourna of physca scences 7(/):5-4. [6] Kangogo,. 8. Some propertes of symmetrc group S6 and assocated combnatora formuas and structures,.sc. Proect; Kenyatta, Unversty, Kenya. [7] Krshnamurthy, V Combnatorcs, theory and appcatons, Affated East West Press Prvate Lmted, New Deh. [8] Neumann, P Fnte permutaton groups edge-cooured graphs and matrces edted by. P. J. Curran, Academc Press, London. [9] Petersen, J Sur e. Theore me de Tat Intermed ath 5:5-7. [] Rose, J. S A Course on group theory. CambrdgeUnversty Press, Cambrdge. [] Rosen, K. H Handbook of Dscrete and combnatona athematcs. CRC Press, New Jersey. [] Rotman, J. J The theory of groups: An ntroducton. Ayn and Bacon, Inc. Boston, U.S.A. [3] Rotch, K. S. 8. Some propertes of the symmetrc group S7 actng on ordered and unordered Pars and the assocated combnatora structures,.sc. Proect, Kenyatta Unversty,Kenya. [4] Gaos, C. 83. Sef-Conugate sub-groups; smpe and composte groups. CambrdgeUnversty Press Cambrdge. [5] Jordan, C. 87. Trate des substtutons et des Equatons Agebrques. Gauther Vas, Pars. [6] Sms, C. C Graphs and fnte permutaton groups. ath. Zetschrft 95: [7] Weandt, H Fnte Permutaton Groups. Academc Press, New York and London. APPENDI:. NOTATIONS ). S, Symmetrc group of degree n and order n!. n ). G, The order of a group G. )., the set of unordered pars from the set,,..., n v). tq,, Unordered par. v). v). v)., the th orbt or suborbt., the ntersecton number reatve to a suborbt a., ntersecton matrx of a suborbt a.. DEFINITION AND PRELIINARY RESULTS Defnton The aternatng group A n s the subgroup of S n comprsng of a n! even permutatons.its order s. Defnton Let be a fnte set,, n, then a symmetrc group of reguar n-gon s caed a dhedra group denoted by. Defnton 3 When G act on a set, s dvded nto dsoned equvaence casses of the acton caed orbts. The orbts contanng s caed the orbt of x, denoted by Orb ( x ). Defnton 4 Let G be transtve on and et G x be the stabzer of a pont x. The orbts,,,... r of G x on are the suborbts of G. Defnton 5 The rank r of G s the number of the suborbts of G whe the engths of the suborbts of G are known as the subdgrees of G. Note: The cardnates of the suborbts and rare ndependent of the choce of x. INTERSECTION NUBERS AND INTERSECTION ATRICES Defnton 6 Let G be a fnte group actng on a fnte set and G n a be the th G a -orbt for a and for a gven arrangement of the G a -orbts. The G b -orbt, b, are aso arranged such the b, then g a g a g a b. The ntersecton numbers reatve to a suborbt a are defned by b I a, b a Defnton 7 If the rank of G s r, then the r r matrx s, IJSTR 4 6

7 INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUE 3, ISSUE, DECEBER 4 ISSN caed the ntersecton matrx of a Theorem [Hgman, []] n and s the suborbt pared wth If a) b) c) n f f f f n n and n n n., then IJSTR 4 7