1-Designs from the group $PSL_{2}(59)$ and their automorphism groups

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1-Designs from the Group

P SL

2

(59)

and

their Automorphism Groups

Reza Kahkeshani

?

Abstract

In this paper, we consider the projective special linear groupP SL2(59)

and construct some 1-designs by applying the Key-Moori method on P SL2(59). Moreover, we obtain parameters of these designs and their auto-morphism groups. It is shown thatP SL2(59)and P SL2(59) : 2appear as

the automorphism group of the constructed designs.

Keywords: Design, automorphism group, projective special linear group.

2010 Mathematics Subject Classification: Primary 05E15, Secondary 05E20, 05B05.

How to cite this article

R. Kahkeshani, 1-Designs from the groupP SL2(59)and their

autom-orphism groups,Math. Interdisc. Res. 3(2018) 147 - 158.

1. Introduction

In [8], Key and Moori considered the primitive actions of the Janko groups J1

andJ2 and construct some designs, codes and graphs. They proved thatJ1 and

J2 apear as the automorphism group of thees combinatorial objects. Key et al.

[10] applied the same method to the groups P SPn(q), A6 ∼= P SL2(9) and A9

and their aim was to construct designs Dfrom a groupGsuch that Aut(D)and Aut(G)have no containment relationship. Motivated by the method used in [8, 9], Darafsheh et al. [5, 6] considered all the primitive actions of the groupsP SL2(q),

where q = 8,11,13,16,17,19,23,25,27,29,31,32, and found the parameters of the obtained designs and determined their automorphism groups. In following, Darafsheh et al. [7] considered the designs constructed by the action of the groups

?_{Corresponding author (E-mail: kahkeshanireza@kashanu.ac.ir)}

Academic Editor: Mohammad Reza Darafsheh Received 23 November 2016, Accepted 29 March 2018 DOI: 10.22052/mir.2017.68740.1048

c

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P SL2(q), where q = 37,41,43,47,49, and obtain parameters and automorphism

groups for all the designs. Moreover, Darafsheh [4] considered the groupP SL2(q),

whereqis a power of2, and found a certain 1-designDinvariant under the group P SL2(q)such thatAut(D)∼=Sq+1.

In this paper, we consider the designs obtained by the primitive permutation representations of the group P SL2(59). Moreover, we obtain the automorphism

groups of the constructed designs.

2. Preliminaries

Letpbe a prime number, nbe a positive integer andq=pn_{. Denote by}_F q the Galois field of orderq. LetGL2(q)be the group of all the invertible2×2matrices

a b c d

over the fieldFq. Denote bySL2(q)the subgroup of GL2(q)consisting

of the matrices with determinant 1. There is a natural action of GL2(q) on the

1-dimensional subspaces of the 2-dimensional vector spaceFq2. The kernel of this action isN :={λI|06=λ∈Fq}. The projective general linear groupP GL2(q)is

the quotientGL2(q)/N. The natural map

(

φ: GL2(q)→Fq∗, M 7→detM,

is a group epimorphism withker(φ) =SL2(q), where Fq∗ =Fq \ {0}. The group SL2(q)also acts on the same set with the kernelN∩SL2(q). Now, the projective

special linear group P SL2(q) is defined to be SL2(q)/(N ∩SL2(q)). There is

another approach for the definition of these linear groups. We add a distinguish

symbol∞to Fq and associated to the invertible matrixM =

a b c d

define the

permutation

fM(x) =



   

   

ax+b

cx+d ifx∈Fq andcx+d6= 0, ∞ ifx∈Fq andcx+d= 0,

a

c ifx=∞andc6= 0, ∞ ifx=∞andc= 0,

onFq∪ {∞}. The set of all such permutations is a subgroup ofSq+1isomorphic to

P GL2(q) [12]. Moreover, a subgroup ofP GL2(q) consists of those permutations

fM for whichad−bcis a non-zero square inFq, denoted byL2(q), is isomorphic

toP SL2(q). In other words,

P SL2(q) =

x7→ax+b

cx+d|06=ad−bcis square

.

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(i) A dihedral group of order 2(q−)/d, where d = (2, q−1). Of course, exceptions occur when= 1,q= 3,5,7,9,11and=−1,q= 2,7,9.

(ii) A solvable group of orderq(q−1)/d.

(iii)A4 whenq >3 is a prime number andq≡3,13,27,37 (mod 40).

(iv)S4 whenqis an odd prime number andq≡ ±1 (mod 8).

(v)A5 whenq is of one of the these forms: q= 5mor 4mwhen mis a prime,

qis a prime number congruent to±1 (mod 5), orqis the square of an odd prime number which satisfiesq≡ −1 (mod 5).

(vi)P SL(2, r)whenq=rm_and_m_{is an odd prime number.} (vii)P GL(2, r)whenq=r2_.

For further properties of the groupsP GL2(q)and P SL2(q), we refer the reader

to [3, 13].

Let D= (P,B,I)be an incidence structure, where P andB are respectively point and block sets andIis a subset ofP ×B. For anyp∈PandB∈ B, we write pI B if and only if (p, B)∈ I. We can replace the incidence relation I by the membership relation ∈, i.e. the relation (p, B)∈ I meansp∈B. The incidence structureD= (P,B,I)is called at−(ν, k, λ)design if |P|=ν,|B| =k for any B∈ B and everytpoints ofP is incident with exactlyλblocks ofB. The design Dis called symmetric if the number of points is equal to the number of blocks, i.e. ν =b. Letλs be the number of blocks through any set of spoints, wheres≤t. We know thatλsis independent of the set,

λs=λ

_ν₋_s

t−s

/

_k₋_s

t−s

andDis also ans−(ν, k, λs)design. At−(ν, k, λ)design is called trivial if any subset ofP with cardinality kis a block ofB. In this case,b= ν_k. The dual of the incidence structureD= (P,B,I)isDt_{= (}_B,_P,_I_{). If}_D_{is a}_t₋₍_{ν, k, λ}₎_design thenDt _{is a design with} _b_{points and the block size} _λ

1. The incidence matrix of

D= (P,B,I)is a|B| × |P|matrixAwith entries0or1 whose rows and columns are labeled by blocks inBand points in P such that entry(B, p)∈ B × P is1 if and only ifpis incidence withB. The incidence matrix ofDt _{is the transpose of} A, At_{. Two structures} _D_{= (}_P,_B,_I₎_and _D0 _{= (}_P0_,_B0_,_I0₎_{are called isomorphic} and we writeD ∼=D0 if there is a one to one correspondenceθ:P → P0 such that for allp∈ P and for allB ∈ B:

pI B←→θ(p)I0 θ(B).

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3. Construction

LetG be a permutation group on a set Ωof size n. Suppose that B is a subset ofΩ with|B| ≥2. ThenD= (Ω, BG_,_∈₎_{is an incidence structure, where} _BG ₌ {Bg_|_g_∈_G} _{forms the block set of}_D_{[3]. Moreover, we can see that if the action} of G on Ω is t-homogeneous and |B| ≥ t then D = (Ω, BG_,_∈₎ _{is a} _t₋₍_{ν, k, λ}₎ design with parametersν=n,k=|B|and

λ=b

_k

t

/

_ν

t

= |G| k t

|GB| ν_t

,

whereGBdenotes the stabilizer ofBinGandb=|BG|is the number of all blocks ofD. In particular, when the action ofGonΩis transitive, we obtain a1−(n, k, λ) design, whereλ= [G:GB]k/n. If the action ofGonΩis primitive andB6={ω} is an orbit ofGω onΩthen |GB|=|Gω|. In this case, the constructed designD has parameters 1−(n, k, k)andG acts on it as a group of automorphisms. The method we use in this paper is as follows:

Theorem 3.1. [8, 9] Let G be a finite primitive permutation group acting on a setΩof size n. Letα∈Ωand{α} 6= ∆be an orbit of the stabilizerGα of α. If

B:= ∆G={∆g|g∈G}

and

E :={α, δ}G₌

{α, δ}g_|_g_∈_G

for a given δ ∈ ∆, then the incidence structure D = (Ω,B) forms a symmetric

1−(n,|∆|,|∆|) design. Further, if ∆ is a self-paired orbit ofGα thenΓ = (Ω,E)

is a regular connected graph of valency |∆|, D is self-dual and G acts as an au-tomorphism group on each of these structures, primitive on vertices of the graph and on points and blocks of the design.

Therefore, our procedure for construction of designs according to the construc-tion method outlined in Theorem 3.1 is as follows. LetG be a group andM be a maximal subgroup ofG. TakeΩ be the right cosets ofM in G. Then,G acts primitively on the setΩof sizen. Chooseω ∈Ωand take∆, where|∆|=k >1, be an orbit of the stabilizer Gω on Ω. By Theorem 3.1, ∆G is the block set of a symmetric design D with parameters 1−(n, k, k). If the action of Gon Ω is 2-transitive thenGω, whereω∈Ω, has only two orbitsωandΩ\{ω}onΩ. Hence, the 1-design obtained in this way is trivial.

By [11], we can say that if Dis a design constructed using the above method then

G≤Aut(D). (1)

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conjecture is generally not true and Aut(D) and Aut(G) have no containment relation. In following, the authors [10] considered the simple groupsA6 andA9.

They found two designsD1andD2from the primitive permutation representations

ofA6 andA9, respectively, and showed thatAut(A6)Aut(D1)andAut(D2) Aut(A9). So, it is interesting to construct designs from the primitive actions of a

groupGand then study the containment relation between the groupsG,Aut(G) andAut(D).

Table 1: 1-designs from the groupP SL2(59).

No. Max. Sub. Degree # Length |Aut(D)|

1 D58 1770 46

29(28) 29(1) 58(2) 58(14)

102660 205320 102660 205320

2 A5 1711 38

6 10 12(2) 20(4) 30(5) 60(24)

102660 102660 102660 102660 102660 102660

3 D60 1711 45

15(2) 30(28) 60(14)

102660 102660 205320

4. Designs Constructed from the Group

P SL2

(59)

Using Magma, we can see that the simple groupP SL2(59)is a permutation group

of order102660 = 22_×₃_×₅_×₂₉_×₅₉_{generated by}

(3,54,48,16,5,10,46,23,26,51,30,28,27,38,59,21,50,56,19,49,45,52,34, 8,20,6,22,39,31)(4,25,36,35,33,12,37,40,17,53,58,47,15,9,60,32,24,41 ,57,43,55,29,11,18,14,44,7,13,42)

and

(1,60,2)(3,31,59)(4,20,57)(5,42,58)(6,46,28)(7,26,17)(8,52,39)(9,30, 43)(10,54,23)(11,25,49)(12,44,27)(13,37,51)(14,29,22)(15,38,41)(16,5 6,34)(18,50,35)(19,32,53)(21,24,47)(33,48,40)(36,55,45)

acting on the set{1,2, . . . ,60} of cardinality 60. Up to conjugacy,P SL2(59)has

5 maximal subgroups M1, M2, . . . , M5 of orders 58, 1711, 60, 60 and 60,

respec-tively. By the shape of the maximal subgroups ofP SL2(59), as noted in above,

M1∼=D58,M3∼=M5=∼A5,M4∼=D60andM2is a solvable group. For the

maxi-mal subgroupM2, the number of orbits of the stabilizer is 2 and so the action of

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is trivial and we will not consider it. Moreover, Computations with Magma shows that the maximal subgroupsM3andM5give us the same results. So, we set them

in one row. By Magma, we see that the maximal subgroupMi, wherei= 1,3,5, is generated by the permutationsαiandβiand moreover,M4is generated byα4,

β4,γ4 andδ4 (See Appendix).

Table I contains all the information we obtain about the primitive representa-tions of the groupP SL2(59). In this table, the shapes of the maximal subgroups

are given under the heading ‘Max. Sub.’. The index of a maximal subgroup in P SL2(59) is given under the heading ‘Degree’ and the symbol ‘#’ indicates the

number of orbits of a point stabilizer in the action ofP SL2(59)on the set of right

cosets of a maximal subgroup. The word ‘Length’ denotes the length of the orbit of the stabilizer of a point and an entry m(n) determines n orbits of length m. Also, the heading ‘Aut(D)’ denotes the order of the automorphism group of the obtained designD. All calculations have been carried out using Magma [2] (See Program in Appendix).

Theorem 4.1. (i) The groupP SL2(59)appears as the full automorphism group of

some designs with parameters1−(1770,29,29),1−(1770,58,58),1−(1711,6,6),

1−(1711,10,10), 1−(1711,12,12), 1 −(1711,15,15), 1 −(1711,20,20), 1 − (1711,30,30)and1−(1711,60,60).

(ii) Aut(P SL2(59)) ∼=P SL2(59) : 2 is the full automorphism group of some

designs with parameters1−(1770,29,29),1−(1770,58,58)and1−(1711,60,60). Proof. By Theorem 3.1 and computations with Magma, we obtain the designs which are listed in above.

(i) If D is one of these designs then computations with Magma show that |Aut(D)|=|P SL2(59)|. Now, inequality (1) implies that Aut(D)∼=P SL2(59).

(ii) Computations with Magma show that the automorphism groups of these designs are isomorphic to each other. Denote byDthe design1−(1770,29,29). By Magma, |Aut(D)| = 205320 = 2|P SL2(59)| and there is a maximal subgroup N

ofAut(D)of index 2 such thatN ∼=P SL2(59). Moreover, we find the involution

γ with the cycle type 2870₁30 _in _Aut₍_D₎_\_N _{(See Appendix). This implies that}

Aut(D)∼=P SL2(59) : 2.

Appendix A: Generators

Generators ofM1:

α1=(1,20,29,51,41,46,31,4,10,6,52,49,38,33,60,8,45,37,9,7,28,27,21,

23,58,15,19,47,13)(2,50,16,44,48,5,40,42,36,35,56,54,26,18,55,3,30,2 5,14,11,57,53,59,32,17,22,12,34,43),

β1=(1,55)(2,60)(3,13)(4,42)(5,6)(7,17)(8,43)(9,22)(10,40)(11,58)(12

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).

Generators ofM3:

α3=(1,16,29)(2,48,55)(3,52,43)(4,57,54)(5,36,18)(6,30,60)(7,58,47)(

8,11,10)(9,13,51)(12,23,28)(14,45,32)(15,24,19)(17,33,25)(20,44,50)( 21,34,49)(22,27,38)(26,35,31)(37,41,59)(39,42,40)(46,56,53),

β3=(1,34)(2,54)(3,46)(4,15)(5,11)(6,42)(7,22)(8,56)(9,24)(10,60)(12

,38)(13,44)(14,39)(16,27)(17,52)(18,47)(19,53)(20,26)(21,31)(23,35)( 25,49)(28,45)(29,37)(30,57)(32,48)(33,51)(36,41)(40,58)(43,59)(50,55 ).

Generators ofM4:

α4=(1,19,10,23,17)(2,54,36,33,55)(3,31,45,18,37)(4,6,5,11,49)(7,32,

30,59,20)(8,27,60,14,42)(9,51,43,50,12)(13,38,25,46,15)(16,21,57,58, 53)(22,35,26,44,28)(24,29,52,47,48)(34,40,39,41,56),

β4=(1,42,40)(2,48,30)(3,44,5)(4,18,35)(6,37,26)(7,33,52)(8,39,19)(9

,25,16)(10,27,41)(11,31,28)(12,38,53)(13,58,50)(14,34,17)(15,57,43)( 20,36,29)(21,51,46)(22,49,45)(23,60,56)(24,59,54)(32,55,47),

γ4=(1,4)(2,25)(3,56)(5,23)(6,17)(7,57)(8,22)(9,48)(10,11)(12,24)(13

,36)(14,26)(15,33)(16,30)(18,40)(19,49)(20,58)(21,32)(27,28)(29,50)( 31,41)(34,37)(35,42)(38,54)(39,45)(43,52)(44,60)(46,55)(47,51)(53,59 ),

δ4 =(1,58)(2,31)(3,55)(4,20)(5,32)(6,7)(8,12)(9,27)(10,16)(11,30)(13

,40)(14,43)(15,34)(17,57)(18,36)(19,53)(21,23)(22,24)(25,41)(26,52)( 28,48)(29,35)(33,37)(38,39)(42,50)(44,47)(45,54)(46,56)(49,59)(51,60 ).

The involution in the proof of Theorem 4.1:

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74,1577)(1075,1550)(1076,1373)(1077,1401)(1079,1602)(1081,1314)(1082 ,1672)(1089,1513)(1092,1364)(1094,1556)(1100,1548)(1102,1260)(1104,1 563)(1105,1578)(1106,1407)(1108,1347)(1109,1531)(1110,1543)(1111,125 6)(1112,1297)(1113,1601)(1115,1638)(1123,1430)(1124,1654)(1125,1193) (1126,1220)(1128,1339)(1134,1136)(1143,1657)(1146,1685)(1149,1302)(1 150,1425)(1154,1701)(1161,1387)(1164,1330)(1171,1569)(1173,1382)(117 5,1668)(1176,1667)(1182,1604)(1183,1499)(1188,1514)(1194,1751)(1197, 1427)(1205,1412)(1206,1318)(1209,1739)(1211,1285)(1214,1354)(1221,15 29)(1222,1524)(1225,1374)(1231,1239)(1237,1698)(1240,1753)(1243,1277 )(1246,1679)(1248,1517)(1251,1603)(1252,1404)(1254,1535)(1258,1647)( 1270,1320)(1275,1544)(1281,1293)(1282,1388)(1287,1770)(1288,1648)(12 94,1631)(1301,1408)(1303,1699)(1310,1471)(1313,1389)(1315,1476)(1322 ,1532)(1324,1713)(1325,1453)(1327,1332)(1340,1613)(1341,1361)(1351,1 619)(1355,1528)(1359,1456)(1370,1758)(1377,1730)(1379,1396)(1380,139 7)(1384,1660)(1385,1458)(1414,1692)(1418,1566)(1429,1482)(1434,1516) (1435,1485)(1442,1472)(1449,1575)(1452,1495)(1454,1633)(1463,1761)(1 465,1678)(1473,1705)(1481,1580)(1491,1552)(1506,1723)(1521,1749)(152 5,1663)(1526,1611)(1530,1703)(1533,1635)(1534,1650)(1537,1760)(1545, 1565)(1554,1719)(1557,1690)(1558,1755)(1579,1702)(1582,1605)(1584,16 21)(1590,1639)(1593,1763)(1612,1683)(1620,1623)(1628,1711)(1630,1754 )(1646,1656)(1655,1728)(1676,1757)(1682,1736)(1687,1724).

Appendinx B: A MAGMA Program

//The program, where g=PSL(2,59) and m is one of the its maximal //subgroups

a1,a2,a3:=CosetAction(g,m); st:=Stabilizer(a2,1); orbs:=Orbits(st); "no of orbits=",#orbs; v:=Index(a2,st); "degree",v;

lo:=[#orbs[j]:j in [1..#orbs]]; "seq of orbit length=",lo; for j:=2 to #lo do

"orbs no",j,"of length",#orbs[j]; blox:=Setseq(orbs[j]^a2);

des:=Design<1,v|blox>;

autdes:=AutomorphismGroup(des); "aut des of order",Order(autdes); "---";

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//omitting the trivial designs and the natural representations

Acknowledgement. The author would like to thank the referees for their useful suggestions and comments. This work is partially supported by the University of Kashan under grant number 572764/2.

Conflicts of Interest. The author declares that there is no conflicts of interest regarding the publication of this article.

References

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Reza Kahkeshani

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan,