Q=29 Groups Effect of Types 5D and 5Α on The Points of Projective Plane Over 31,29,F =qq Emad B. A. K. Al-Zangana Department of Mathematics/College of Science/Al-Mustansiriyah University E-mail: emad77_kaka@yahoo.com Abstract The purpose of this paper is to find an arc of degree five in 31,29),(2, =qqPG , with stabilizer group of type dihedral group of degree five 5D and arcs of degree six and ten with stabilizer groups of type alternating group of degree five 5A , then study the effect of 5D and 5A on the points of projective plane. Also, find a pentastigm which has collinear diagonal points. Key words: Projective plane, arc. 410 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 mailto:emad77_kaka@yahoo.com Introduction Let qF denotes the Galois field of q elements and ),3( qV be the vector space of row vectors of length three with entries in qF . Let )(2, qPG be the corresponding projective plane. The points ],,[ 210 xxx of )(2, qPG are the 1-dimensional subspaces of ),3( qV . Subspaces of dimension two of the form )(V 210 cXbXaX ++ are called lines. The number of points and the number of lines in )(2, qPG is 12 ++ qq . There are 1+q points on every line and 1+q lines through every point. The lines of )(2, qPG are constructed by the following way: 1221 ,,, ++ qq iii PPP  be the points lie on the line )V( 21 X= where the index ji refers to the position of the point ji P in the plane. Fixed the line )V( 21 X= as a first line. The second line is constructed by adding 1 to the index ji and so on. Definition 1.1[1]: An );( rn arc K or arc of degree r in ),( qkPG with 1+≥ rn is a set of n points with property that every hyperplane meets K in at most r points of K and there is some hyperplane meeting K in exactly r points. An )2;(n -arc is also called an n -arc. Definition 1.2[1]: An );( rn -arc is complete if it is maximal with respect to inclusion; that is, it is not contained in an );1( rn + -arc. Definition 1.3[1]: A line of ),( qkPG , 1>k is an i -secant of an );( rn arc K if iK = . A 2 -secant is called a bisecant, a 1-secant a unisecant (tangent) and a 0 - secant is an external line. Let ic be the number of points of KqPG \),2( with index exactly i . So, the parameters 0c is the number of points through which no bisecant of K passes and 3c is the number of points where three bisecants meet. Theorem 1.4[1]: ( The Fundamental Theorem of Projective Geometry) If },,{ 10 +nPP  and },,{ 10 +′′ nPP  are both subsets of ),( qnPG of cardinality 2+n such that no 1+n points chosen from the same set lie in a hyperplane, then there exists a unique projectivity ℑ such that ℑ=′ ii PP for .1,,1,0 += ni  Definition 1.5[1]: Let ].1,1,1[U],1,0,0[U],0,1,0[U],0,0,1[U 210 ==== The set }U,U,U,U{ 210=Γq is called the standard frame. Definition 1.6[3]: Let 1χ and 2χ be two projective spaces of dimension n . A projectivity 21: χχ →ℑ is a bijection given by a non-singular 1)(1)( +×+ nn matrix A such that ℑ=′ )P( )P( XX if and only if XA. Xt =′ , where {0}.\F qt ∈ Two projective spaces 1χ and 2χ are projectively equivalent if there is a projectivity between them. Definition 1.7[1]: A point of index three is called a Brianchon point or B -point for short. Write nmlkji PPPPPPmnklj =⋅⋅ for B -point. 411 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Remark 1.8[1]: (i) An );1( rn + arc is constructing from an );( rn -arc, K by adding one point of index zero to K . (ii) Two arcs K and K ′ are projectively equivalent if there is a projectivity between them. (ii) With parameters 0c , K is complete if and only if 00 =c . Some groups that occur in this paper are listed below. For more details see [2]. nC = cyclic group of order n ; nS = symmetric group of degree n ; nΑ = alternating group of degree n ; nD = dihedral group of order n2 1)(, 22 ==== rssrsr n . During this research the primitive element 2=υ in 29F and 3=ω in 31F are used. Hirschfeld stated in [1] that in )9(2,PG there is a unique arc of degree five with stabilizer group isomorphic to a dihedral group of degree five 5D . Also, there is a unique arc of degree six with stabilizer group isomorphic to alternating group of degree five 5A . In 1984, Sadeh [3] and in 2011 Al-Zangana [4] proved the same results in )11,(2PG and )19,(2PG also they proved that the same arc of degree six has ten B -points which is an arc of degree ten with stabilizer group of type 5A . In 1995, Storm and Maldeghem [5] proved in )(2, qPG when 1±≡q )10mod( they proved theoretically these results. According to these previous results, the following question arises: 1- What is the arc of degree five which has stabilizer group of type dihedral group of degree five 5D and what is the effect of 5D on the points of projective plane. 2- What is the pentastigm which has collinear diagonal points. 3- What is the arcs of degree six and ten which have stabilizer groups of type alternating group of degree five 5A and what is the effect of 5A on the points of projective plane. 1- Inequivalent Arcs of Degree Five From the fundamental theorem of Projective Geometry, there is a unique inequivalent arc of degree four in the projective plane with stabilizer group isomorphic to 4S . So, the standard frame }U,U,U,U{ 210=Γq formed a projectively unique 4 -arc in the projective plane. An arc of degree five is constructed by adding one point of index zero to qΓ . And to find equivalents arcs, a mathematical programming language GAP has been used [6]. Theorem 2.1: (i) There are ten inequivalent 5-arcs through the frame 29Γ in )29,2(PG with parameters ]15,250,601[],,[ 210 =ccc as summarized in Table 1. (ii) There are eleven inequivalent 5-arcs through the frame 31Γ in )31,2(PG with parameters ]15,270,703[],,[ 210 =ccc as summarized in Table 2. Table 1: Inequivalent 5-arc and their stabilizer group in (2,29)PG No. 5-Arc Stabilizer Group 1 },1),P({ 1427291 vvA Γ= ] ] 1, , [ ], 0 0, , [ ], 0 , 0, [ [C 14131315 2 vvvv= 2 },1),P({ 1124292 vvA Γ= I 412 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 3 },1),P({ 62293 vvA Γ= ] ] 0 , 0, [ ], 0, 0, [ ], 0 0, , [ [C 201417 2 vvv= 4 },1),P({ 48294 vvA Γ= ] ] 0 0, , [ ], 0 , 0, [ ], 0, 0, [ [C 221814 2 vvv= 5 },1),P({ 1210295 vvA Γ= I 6 },1),P({ 1612296 vvA Γ= ] ] 0, 0, [ ], 0 0, , [ ], 0 , 0, [ [C 14262 2 vvv= 7 },1),P({ 1526297 vvA Γ= ] ] 0, 0, [ ], , , [ ], , , [ [C 32626261412 2 vvvvvvv= 8 },1),P({ 527298 vvA Γ= I 9 },1),P({ 718299 vvA Γ= ] ] 0 0, , [ ], 1 1, 1, [ ], 0 , 0, [ [C 1414 4 vv= 10 },1),P({ 16222910 vvA Γ= ] ] 0 0, , [ ], 1 1, 1, [ ], 0 , 0, [ [ ], ] 0 , 0, [ ], 0, 0, [ ], 0 0, , [ [ D 28 2148 5 vvs vvvr = = = Table 2: Inequivalent 5-arc and their stabilizer group in (2,31)PG No. 5-Arc Stabilizer Group 1 },1),P({ 102311 ωωΓ=′A I 2 },1),P({ 46312 ωωΓ=′A I 3 },1),P({ 2712313 ωωΓ=′A ] ] , , [ ], , , [ ], 0 0, , [ [C 33315182719 2 ωωωωωωω= 4 },1),P({ 2913314 ωωΓ=′A I 5 },1),P({ 1117315 ωωΓ=′A I 6 },1),P({ 2014316 ωωΓ=′A ] ] ,0 ,0 [ ], , , [ ], , , [ [C 1714141415529 2 ωωωωωωω= 7 },1),P({ 915317 ωωΓ=′A ] ] 0 0,1, [ ], 1 ,0 ,0 [ ], , , [ [C 151515 2 ωωω= 8 },1),P({ 208318 ωωΓ=′A ] ] ,,[ ], ,0 ,0 [ ], , , [ [ 151515615523 2 ωωωωωωω=C 9 },1),P({ 926319 ωωΓ=′A ] ] 0, ,0 [ ], , , [ ], , , [ [C 152411999 4 ωωωωωωω= 10 },1),P({ 2553110 ωωΓ=′A ] ] 1 1, ,1 [],0 ,0, [ ], 0 ,0, [ [ ], ] ,0 0, [ ],0 0, , [ ], 0 , ,0 [ [ S 2010 152010 3 ωω ωωω = 11 },1),P({ 2743111 ωωΓ=′A ] ] 1 1, ,1 [],, , [ ], 0 ,0, [ [ ], ] , , [ ],1 1, ,1 [ ], 0 ,0 , [ [ D 1511197 15111926 5 ωωωω ωωωω =′ =′ = s r Collinearities of the Diagonal Points of Pentastigm. Definition 3.1[3]: An n -stigm in ),2( qPG is a set of n points, no three of which are collinear, together with the )1( 2 1 −nn lines that are joins of pairs of the points. The points and lines are called vertices and sides of the n -stigm. The vertices form an n -arc. A 5-stigm is also called pentastigm. The diagonal points of an n -stigm are the intersections of two sides which do not pass through the same vertex. In general, any 5-arc has 15 diagonal points since 413 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ,15 2 3 2 5 2 1 =            and these points are exactly the fifteen points of index two. Write klij ⋅ for lkji PPPP  . Let ),,P(,U,U,U,U 21043221100 aaaPPPPP ===== be the vertices of a pentastigm ρ . So, the following equation is satisfied: .0))()(( 212010210 ≠−−− aaaaaaaaa In this section, the inequivalent 5-arc that has a five diagonal points is found in ),2( qPG , .31,29=q Lemma 3.2[1]: The condition that five diagonal points of a pentastigm ρ are collinear in ),2( qPG is that 012 =−− xx has solution in qF . Corollary 3.3: (i) If ,29=q the equation 012 =−− xx has two solutions 5,6 − . (ii) If ,31=q the equation 012 =−− xx has two solutions 12,13 − . So, there is a Pentastigm with five collinear points in )29,2(PG and )31,2(PG . Theorem 3.4: (i) In )29,2(PG , the pentastigm which has the 5-arc 10A as vertices has five diagonal points which are collinear on the line )5V( 210627 XXX −−= as shown below. ).1,,(P2314)15(),1,,0(P1204)10(),1,0,(P1402)5( ),1,,1(P2413)14(),1,1,(P2403)9(),1,0,1(P1302)4( ),1,,0(P3412)13(),1,1,(P1403)8(),0,1,(P3401)3( ),1,,(P2304)12(),1,1,0(P1203)7(),0,1,(P2401)2( ),1,,1(P1304)11(),1,0,(P3402)6(),0,1,1(P2301)1( 22221622 226 82222 16166 1616 vvvv vv vvv vvv vv =⋅=⋅=⋅ =⋅=⋅=⋅ =⋅=⋅=⋅ =⋅=⋅=⋅ =⋅=⋅=⋅ Amongst these diagonal points, the five diagonal points ),1,,0(P3412 ),1,,1(P1304 ),1,1,(P2403 ),1,0,(P1402 ),0,1,1(P2301 8 16 6 22 v v v v =⋅ =⋅ =⋅ =⋅ =⋅ 414 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 lie on the line )5V( 210627 XXX −−= . (ii) In )31,2(PG , the pentastigm which has the 5-arc 11A′ as vertices has five diagonal points which are collinear on the line )1219V( 210379 XXX ++= as shown below. ).1,,(P2314)15(),1,,0(P1204)10(),1,0,(P1402)5( ),1,,1(P2413)14(),1,1,(P2403)9(),1,0,1(P1302)4( ),1,,0(P3412)13(),1,1,(P1403)8(),0,1,(P3401)3( ),1,,(P2304)12(),1,1,0(P1203)7(),0,1,(P2401)2( ),1,,1(P1304)11(),1,0,(P3402)6(),0,1,1(P2301)1( 44264 228 4419 26268 2625 ωωωω ωω ωωω ωωω ωω =⋅=⋅=⋅ =⋅=⋅=⋅ =⋅=⋅=⋅ =⋅=⋅=⋅ =⋅=⋅=⋅ Amongst these diagonal points, the five diagonal points ),1,,1(P2413 ),1,,(P2304 ),1,0,0(P1203 ),1,0,(P1402 ),0,1,(P3401 22 2626 4 19 ω ωω ω ω =⋅ =⋅ =⋅ =⋅ =⋅ lie on the line )1219V( 210379 XXX ++= . The Group Action of 5D on the 5-Arc In this section, the group action of 5D on the 5-arc 10A in )29,2(PG and on the 5-arc 11A′ in )31,2(PG has been studied. (I) When 29=q . From Table 1, the Dihedral group 5D generated by           =           = 00 111 00 , 00 00 00 2 8 2 14 8 v v s v v v r is the stabilizer group of the 5 -arc 10A . The effects of the group 5D on the projective plane )29,2(PG are given below. 1. The group 5D fixes the conic 10AC . 2. The group 5D acts transitively on 10A since 10 U),U( rs , 2 4 0 U),U( rs , 415 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 U),U( 20 rs , ,1),P(),U( 162230 vvrs  . 3. Each of the five projectivities 432 ,,,, rsrsrsrsr fixes 25 points amongst the 601 points of index zero by transforming each point to itself. 4. Each of these 25 points lies on a line which is a unisecant to 10A and a bisecant of the conic 212010 6510 XXXXXXC A −+= . These lines are );V(5 12244 XX −= );V( 210790 XXX −+= );4V( 20659 XX += );5V( 1072 XX −= ).56V( 210422 XXX +−= 5. Each of the five projectivities 432 ,,,, rsrsrsrsr fixes 31 points of )29,2(PG by transforming each point to itself. These points are exactly the following: ,1)};P(0,{ 8244 v ,0)};1 P(1,{790  ,1,1)};P({ 6659 v ,1)};P(1,{ 1672 v Table 3: Points of index zero fixed by elements of 5D in )29,2(PG },,,,{\ 54321 PPPPPi },1),P(,1),,0P(,1),,P(,,1)P(0,,P(1,0,0){\ 222222221722244 νννννν r 1 },1,1)0P(,1),,P(,0,1),1P(,,1),P(,,1,0)P({\ 2727162214790 ννωων rs 2 },1),P(,1),1,P(,1),,P(,,1)0,P(,P(0,1,0){\ 101616161616659 νννννν 2rs 3 },1,1)P(,1,0),P(,1),,1P(,,1),P(,,0,1)0P({\ 22226161072 ννννν 3rs 4 },0,1)P(,1),,P(,1,0),P(,,1),0P(,,1)1,1P({\ 81722616422 ννννν 4rs 5 6. These additional points to the lines are exactly the diagonal points of 5-arc 10A . 7. The other four projectivities fix only one point )1,,(P 2216 vv which is the point intersection of the five lines 244 , 790 , 659 , 72 , 422 . (II) When 31=q . From Table 2, the Dihedral group 5D generated by           =′           =′ 111 00 ,111 00 151119 7 151119 26 ωωω ω ωωω ω sr ,0,1)}.P({ 22422 v 416 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 is the stabilizer group of the 5 -arc 11A′ . The effects of the group 5D on the projective plane )31,2(PG are given below. 1. The group 5D fixes the conic 11AC ′ . 2. The group 5D acts transitively on 11A′ since 10 U),U( sr ′′ , 2 3 0 U),U( sr ′′ , U),U( 40 sr ′′ , ,1),P(),U( 27420 ωωs′ . 3. Each of the five projectivities 432 ,,,, srsrsrsrr ′′′′′′′′′ fixes 27 points amongst the 703 points of index zero by transforming each point to itself. 4. Each of these 27 points lies on a line which is a unisecant to 11A′ and a bisecant of the conic }F |t)),9(15),77(P({1817 *31 2 21201011 ∈−−−−=−+=′ ttttXXXXXXC A . These lines are );V(19 12643 XX −=′ );2V(5 10927 XX −=′ );11V( 2029 XX +=′ );1911V( 210900 XXX ++=′ ).V( 210757 XXX −−=′ Table 4: Points of index zero fixed by elements of 5D in )31,2(PG },,,,{\ 54321 PPPPPi′ },1)P(1,,1),,P(,1),,P(,,1)P(0,,P(1,0,0){\ 4412444643 ωωωωωω′ r′ 1 },1,1)P(,1),,1P(,0),1,P(,,1),P(,P(0,0,1){\ 42641014927 ωωωωω′ sr ′′ 2 },0,1)P(,1),,P(,1),1,P(,,1),P(,P(0,1,0){\ 826888829 ωωωωωω′ 2sr ′′ 3 },1)P(0,,0,1),P(,1),,P(,,1)1,1P(,,1,0)P({\ 26192048900 ωωωωω′ 3sr ′′ 4 },1,1)P(,1,0),1P(,0,1),1P(,,1),0P(,,1),P({\ 2415264757 ωωωω′ 4sr ′′ 5 5. Each of the five projectivities 432 ,,,, srsrsrsrr ′′′′′′′′′ fixes 33 points of )31,2(PG by transforming each point to itself. These points are exactly the following: ,1)};P(1,{ 22643 ω′ ,0)};1 ,P({ 19927 ω′ ,1)};,P({ 262629 ωω′ ,0,1)};P({ 4900 ω′ ,1,1)}.0P({757 ′ 417 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 6. These additional points to the lines are exactly the diagonal points of 5-arc 11A′ . 7. The other four projectivities fix three points ,1),P( 2927 ωω , ,1),P( 233 ωω , ,1),P( 48 ωω which ,1),P( 48 ωω is the intersection point of the five lines 643′ , 927′ , 29′ , 900′ , 757′ . Unique Inequivalent Arc of Degree Six With Stabilizer of Type 5A In this section, the unique 6-arc Κ through the frame which has a stabilizer group )(KG isomorphic to 5A is found. Also, the effect of 5A on the 31,29),,2( =qqPG is discussed. Let )}.1,,(P),1,,(P,U,U,U,U{ 654231201 dcPbaPPPPPK ======= There are fifteen ways of choosing three bisecants no two of which intersect on Κ . These three bisecants form either a triangle or will intersect at a B -point. (I) When 29=q . From the 5-arc 10A , an arc of degree six 29β is constructed by adding )1,,(P 814 vv of index zero; that is, )}.1,,(P{ 8141029 vvA =β This arc has the following properties: 1- This arc has parameters 10]. 15, 360, 480, [],,,[ 3210 =cccc Since 00 ≠c , so 29β is not complete arc. 2- The stabilizer group of 29β is of type 5A as given below 1)(|,)( 53229 ==== ghhghgG β , where           =           = 00 100 00 , 111 00 00 16 22 14 14 v v hv v g . 3- The group )( 29βG has a subgroup of type 5D generated by 21 ,αα fixes the conic }F |)9),13(1),P(4({65 *29 2 21201010 ∈−+−+=−+= tttttXXXXXXC A , where .1, 00 111 00 , 00 00 00 5 2 2 1 2 8 2 2 14 8 1 ==           =           = αααα v v v v v 4-The calculation shows that the parameter 103 =c , so 29β has ten B -Points as given below. 418 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ,1).,P(342516(10),0,1);0P( 5623 14(5) ,1);,1P(35 42 16(9),0,1);P( 45 6 2 13(4) ,1);,P( 342615(8),0,1);1P( 562413(3) ,1);,0P( 462315(7),1,0);P( 4536 12 (2) ,1,1);P( 362514)6(,1,0);P( 6 4 35 2 1(1) 2222 2216 1616 1622 226 vv vv vv vv vv =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ 5-The set ,1),,0P(,1,1),P(,0,1),0P(,0,1),P(,0,1),1P(,1,0),P(,1,0),P({ 162216226 vvvvv=Κ },1),P(,1),,1P(,1),,P( 2222221616 vvvvv of B -Points of 29β form a non complete 10-arc with parameters ].6,15,90,210,480,60[],,,,,[ 543210 =cccccc The stabilizer group of K is isomorphic to 5A . 8. The remaining five possibilities form triangles. In Table 5, the sides of these triangles and their vertices are given. Table 5: Five triangles fixed by 5A in )29,2(PG (I) (II) (III) (IV) (V) )(V 221 XPP = )(V 1043 XXPP −= )(V 21065 XXXPP −+= )(V 131 XPP = )5(V 2052 XXPP −= )56(V 21064 XXXPP +−= )(V 2141 XXPP −= )4(V 2062 XXPP += )5(V 1053 XXPP −= )4(V 2151 XXPP += )(V 2042 XXPP −= )5(V 1063 XXPP −= )5(V 1261 XXPP −= )(V 032 XPP = )56(V 21054 XXXPP −−= ,1,0)1P( ,1)1,P( 14v ,1),P( 2727 vv ,0,1)P( 22v ,1)0,P( 8v ,1),P( 1722 vv ,1,1)(P 16v ,1,1)P( 6v ,1),P( 1016 vv ,1),1P( 16v ,1),P( 1610 vv ,1),1P( 6v ,1),0P( 22v ,1),(P 2217 vv ,1),0P( 8v 7- Let }VIV,III,II,I,{=W be the set of five triangles in Table 5. Each elements of the group 5A≅)( 29βG fixes the set W . (II) When 31=q . From the 5-arc 11A′ , an arc of degree six 31β is constructed by adding ,1),P( 48 ωω of index zero; that is, )}.1,,(P{ 481131 ωωβ A′= This arc has the following properties: 1- 31β has parameters 10]. 15, 390, 572, [],,,[ 3210 =cccc Since 00 ≠c , so 31β is not complete arc. 2- The stabilizer group of 31β is of type 5A as given below 419 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 1)(|,)( 53231 =′′=′=′′′= hghghgG β , where           =′           =′ 00 100 00 , 111 00 00 10 20 16 16 ω ω ω ω hg . 3-The identity subgroup of the group )( 31βG fixes the conic 212010 181711 XXXXXXC A −+=′ . 4-The calculation show that the parameter 103 =c , so 31β has ten B -Points as given below. ,1).,P(342516(10),1,1);P( 3626 14(5) ,1);,0P(45 32 16(9),0,1);P( 45 6 2 13(4) ,1);,1P( 362415(8),0,1);1P( 562413(3) ,1);,0P( 462315(7),1,0);P( 4635 12 (2) ,1,1);P( 352614)6(,1,0);1P( 6 5 34 2 1(1) 444 48 26 268 8 ωωω ωω ω ωω ω =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ =⋅⋅=⋅⋅ 5-The set ,1,1),P(,1,1),P(,0,1),P(,0,1),1P(,1,0),P(,1,0),1P( { 8488 ωωωω=Κ′ },1),P(,1),,0P(,1),,1P(,1),,0P( 4442626 ωωωωω of B -Points of 31β form a non complete 10- arc with parameters ].6,15,90,210,480,60[],,,,,[ 543210 =cccccc The stabilizer group of K′ is isomorphic to 5A . 8. The remaining five possibilities form triangles. In Table 6, the sides of these triangles and their vertices are given. Table 6: Five triangles fixed by 5A in )31,2(PG (I) (II) (III) (IV) (V) )(V 221 XPP = )12(V 1063 XXPP += )1112(V 21054 XXXPP +−= )(V 131 XPP = )19(V 2052 XXPP −= )1911(V 21064 XXXPP ++= )(V 2141 XXPP −= )(V 033 XPP = )(V 21065 XXXPP −−= )18(V 1251 XXPP −= )20(V 2062 XXPP −= )(V 1043 XXPP −= )19(V 1261 XXPP −= )(V 2042 XXPP −= )11(V 1053 XXPP += 420 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 ,1,0)P( 4ω ,1)0,P( 19ω ,1),P( 1014 ωω ,0,1)P( 4ω ,1)0,P( 19ω ,1),P( 204 ωω ,1,1)0(P ,1,1)P( 24ω ,1),0P( 15ω ,1),P( 268 ωω ,1),P( 2626 ωω ,1),P( 88 ωω ,1),1P( 4ω ,1),(P 412 ωω ,1),1P( 22ω 9. Let }VIV,III,II,I,{=′W be the set of five triangles in Table 6. Each elements of the group 5A≅)( 31βG fixes the set W ′. Conclusion 1- There is an arc of degree five },,,,{ 54321 PPPPP=ξ which has stabilizer group )(ξG of type 5D . 2- The pentastigm which has ξ as a vertex has collinear diagonal points. 3- The effect of the group )(ξG on points of 31,29),,2( =qqPG depends on the order of its elements. Let 2G be the set of five elements of )(ξG of order two and 5G be the set of four elements of )(ξG of order five. (i) Each element of 2G fixes five a subset of the plane of length 2+q by sending it to itself. Each of this set, is a line *i with extra point * iP , 5,4,3,2,1=i . The five extra points iP′ are exactly the diagonal points of ξ . Also, these lines are the bisecant to the conic ξC which passes through ξ and unisecants to ξ . (ii) Each element of 5G fixes a point ∗P which is the intersection point of the five lines .5,4,3,2,1,* =ii 4- The unique six arc with stabilizer group of type 5A is constructed by adding the point ∗P to ξ . So, the following figure is fixed by the group )(ξG . 421 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 Figure 1. According to these results and Storm and Maldeghem results [4, proposition 12] the following conjecture is deduced. Conjecture: In )(2, qPG , when 1±≡q )10mod( there is a unique arc of degree five ξ fixed by group )(ξG of type 5D and there is a unique arc of degree six consists of ξ and a point ∗P which is fixed by the elements of )(ξG of degree five. And the group )(ξG fixed the Figure 1. References [1] Hirschfeld J. W. P., (1998), Projective geometries over finite fields, 2nd Edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York. [2] Thomas A. D. and Wood G. V., (1980), Group tables, Shiva Mathematics Series, Series 2., Devon Print Group, Exeter, Devon, UK. [3] Sadeh A. R., (1984), Cubics surfaces with twenty seven lines over the eleven elements, PhD thesis, University of Sussex, United Kingdom. [4] Al-Zangana E. M., (2011), The geometry of the plane of order nineteen and its application to error-correcting codes, Ph.D. thesis, University of Sussex, United Kingdom. [5] Storme L. and Maldeghem V., (1995), Primitive arcs in ),2( qPG , J. Combin. Theory, Ser. A, 69, (200-216). [6] GAP Group, (2013), GAP. Reference manual, URL http://www.gap-system.org. 422 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013 http://www.gap-system/ ، qF فيالمستوي األسقاطي نقاط في 5Aو 5Dواع تاثیر الزمر من األن [ ]31,29=q عماد بكر عبد الكریم الزنكنة الجامعة المستنصریة / كلیة العلوم /قسم الریاضیات الخالصة 31,29,,2),(في المستوي األسقاطي قوس من الدرجة الخامسة یجادأرض من ھذا البحث لغا qPGq ذات ، = و اقواس من الدرجة السادسة والعاشرة ذات زمرة مثبتة 5Dزمرة داھیدرل من الدرجة الخامسة زمرة مثبتة من النوع سقاطي . نقاط المستوي األ في 5Aو 5Dومن ثم دراسة تأثیر 5Aنوع الزمرة المتناوبة من الدرجة الخامسة من قامة واحدة.توكذلك، ایجاد بنتاستام ذات نقاط قطریة على اس الكلمات المفتاحیة: المستوي االسقاطي، القوس. 423 | Mathematics @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹1a26@@ÖÜ»€a@I3@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (3) 2013