Research article

On the symmetric block design with parameters (280,63,14) admitting a Frobenius group of order 93

  • Received: 07 April 2019 Accepted: 14 August 2019 Published: 02 September 2019
  • MSC : 05B05

  • In this paper we have proved that for a putative symmetric block design D with parameters (280, 63, 14), admitting a Frobenius group G=ρ,μ|ρ31=μ3=1,ρμ=ρ5 of order 93, there are exactly thirteen possible orbit structure up to isomorphism; two with the orbit distribution [1;31;31;31;93;93], eight with the orbit distribution [1;31;31;31;31;31;31;93] and three with the orbit distribution [1;31;31;31;31;31;31;31;31;,31].

    Citation: Menderes Gashi. On the symmetric block design with parameters (280,63,14) admitting a Frobenius group of order 93[J]. AIMS Mathematics, 2019, 4(4): 1258-1273. doi: 10.3934/math.2019.4.1258

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  • In this paper we have proved that for a putative symmetric block design D with parameters (280, 63, 14), admitting a Frobenius group G=ρ,μ|ρ31=μ3=1,ρμ=ρ5 of order 93, there are exactly thirteen possible orbit structure up to isomorphism; two with the orbit distribution [1;31;31;31;93;93], eight with the orbit distribution [1;31;31;31;31;31;31;93] and three with the orbit distribution [1;31;31;31;31;31;31;31;31;,31].


    A 2(v,k,λ) design (P,B,I) is said to be symmetric if the relation |P|=|B|=v holds and in that case we often speak of a symmetric design with parameters (v,k,λ). The collection of the parameter sets (v,k,λ) for which a symmetric 2(v,k,λ) design exists is often called the "spectrum". The determination of the spectrum for symmetric designs is a widely open problem. For example, a finite projective plane of order n is a symmetric design with parameters (n2+n+1,n+1,1) and it is still unknown whether finite projective planes of non-prime-power order may exist at all.

    The existence/non-existence of a symmetric design has often required "ad hoc" treatments even for a single parameter set (v,k,λ). The most famous instance of this circumstance is perhaps the non-existence of the projective plane of order 10, see [11].

    It is of interest to study symmetric designs with additional properties, which often involve the assumption that a non-trivial automorphism group acts on the design under consideration, see for instance [4].

    Among symmetric block designs of square order, a study of symmetric block designs of order 49 is of a particular interest. There are 15 possible parameters (v,k,λ) for symmetric designs of order 49, but until now only a few results are known (see [5,8]). Due to the fact that symmetric designs of order 49 have a big number of points (blocks), the study of sporadic cases is very difficult, except, possibly, when the existence of a collineation group is assumed.

    A few methods for the construction of symmetric designs are known and all of them have shown to be effective in certain situations. Here, we shall use the method of tactical decompositions, assuming that a certain automorphism group acts on the design we want to construct, used by Z. Janko in [9]; see also [4,10] and [6]. The present paper is concerned with a symmetric design D=(P,B,I) with parameters (280,63,14): the existence/non-existence of such a design is still in doubt as far as we know. We shall further assume that the given design admits a certain automorphism group of order 93. We assume the reader is familiar with the basic facts of design theory, see for instance [2], [3] and [12]. If g is an automorphism of a symmetric design D with parameters (v,k,λ), then g fixes an equal number of points and blocks, see [12,Theorem 3.1,p.78]. We denote the sets of these fixed elements by FP(g) and FB(g) respectively, and their cardinality simply by |F(g)|. We shall make use of the following upper bound for the number of fixed points, see [12,Corollary 3.7,p. 82]:

    |F(g)|k+kλ. (1)

    It is also known that an automorphism group G of a symmetric design has the same number of orbits on the set of points P as on the set of blocks B: [12,Theorem 3.3,p.79]. Denote that number by t.

    We adopt the notation and terminology of Section 1 in [4]. In the following, for the sake of completeness, some fundamental relations are explicitly provided. Let D be a symmetric design with parameters (v,k,λ) and let G be a subetaoup of the automorphism group Aut(D) of D. Denote the point orbits of G on P by P1,P2,Pt and the line orbits of G on B by B1,B2,Bt. Put |Pr|=ωr and |Bi|=Ωi. Obviously,

    tr=1ωr=ti=1Ωi=v. (2)

    Let γir be the number of points from Pr, which lie on a line from Bi; clearly this number does not depend on the chosen line. Similarly, let Γjs be the number of lines from Bj which pass through a point from Ps. Then, obviously,

    tr=1γir=k and tj=1Γjs=k. (3)

    By [3,Lemma 5.3.1. p.221], the partition of the point set P and of the block set B forms a tactical decomposition of the design D in the sense of [3,p.210]. Thus, the following equations hold:

    Ωiγir=ωrΓir (4)
    tr=1γirΓjr=λΩj+δij(kλ) (5)
    ti=1Γirγis=λωs+δrs(kλ) (6)

    where δij, δrs are the Kronecker symbols.

    For a proof of these equations, the reader is referred to [3] and [4]. Equation (5), together with (4) yields

    tr=1Ωjωrγirγjr=λΩj+δij(kλ). (7)

    Definition 1. We denote

    [Li,Lj]=tr=1Ωjωrγirγjr,1i,jt

    and call these expressions the orbit products. The (t×t)-matrix (γir) is called the orbit structure of the design D.

    The first step in the construction of a design is to find all possible orbit structures. The second step of the construction is usually called indexing. In fact for each coefficient γir of the orbit matrix one has to specify which γir points of the point orbit Pr lie on the lines of the block orbit Bi. Of course, it is enough to do this for a representative of each block orbit, as the other lines of that orbit can be obtained by producing all G-images of the given representative.

    In our construction of symmetric 2(280,63,14) designs we assume the existence of an automorphism group G=ρ,μ|ρ31=μ3=1,ρμ=ρ5, which is a so called Frobenius group of order 93 with Frobenius kernel of order 31 (see [7]).

    Lemma 3.1. Let ρ be an element of G with o(ρ)=31. Then ρ fixes precisely one point and one block.

    Proof. By [12,Theorem 3.1] the group ρ fixes the same number of points and blocks. Denote that number by f. Obviously f280(mod31), i.e. f1(mod31). The upper bound (1) for the number of fixed points yields f{1,32,63}. As o(ρ)>λ, an application of a result of M. Aschbacher [1,Lemma 2.6,p.274] forces the fixed structure to be a subdesign of D. But there is no symmetric design with v=32 or v=63 and λ=14 (there is no kIN which satisfy 14(v1)=k(k1)). Hence, f is equal to 1.

    Our next task is to determine the lengths of the orbits of G on the sets of points and blocks of the symmetric block design D. The possible orbit lengths are 1,3,31,93.

    Lemma 3.2. There is no orbit of length 3 of G.

    Proof. If false, then ρ would have at least three fixed points or three fixed blocks, which is not possible.

    Up to reordering, there are precisely four possibilities for the arrays expressing the lengths of the G-orbits on points and blocks, namely: O1=[1;93;93;93]; O2=[1;31;31;31;93;93]; O3=[1;31;31;31;31;31;31;93];O4=[1;31;31;31;31;31;31;31;31;31]:

    Lemma 3.3. The case O1=[1;93;93;93] of the orbit distribution does not occur.

    Since for the case O1=[1;93;93;93] it is not possible to be constructed the fixed block, the following lemmas follow.

    Lemma 3.4. Up to isomorphism there are exactly two orbit structures for symmetric (280, 63, 14) designs and the automorphism group F313 acting with the orbit distribution O2=[1;31;31;31;93;93].

    Proof. We put PI={I0,I1,,I30},I=1,2,3, P4={40,41,,430,50,51,,530,60,61,,630}, P5={70,71,,730,80,81,,830,90,91,,930}, for the non–trivial orbits of the group G. Thus, G acts on these point orbits as a permutation group in a unique way. Hence, for the two generators of G we may put

    ρ=()(I0,I1,,I30),I=1,2,,9,

    where is the fixed point of collineation, whereas non-trivial ρorbits are numbers 1,2,3,4,5,6,7, 8,9 and ,10,11,,930 all points of the symmetric block design D, and the collineation μ of order 3 acts in the symmetric block design as permutation (1)(2)(3)(4,5,6)(7,8,9) on orbit numbers, whereas on indices acts μ:x5x (mod 31) or

    μ=()(K0)(K1,K5,K25)(K2,K10,K19)(K3,K15,K13)(K4,K20,K7)
    (K6,K30,K26)(K8,K9,K14)(K11,K24,K27)(K12,K29,K21)(K16,K18,K28)
    (K17,K23,K22)(4i,55i,625i)(7i,85i,925i),K=1,2,3,i=0,,30.

    We immediately obtain the following.

    Corollary 1. The element μ of G of order 3 fixes precisely 4 points and 4 blocks of D. Each block orbit contains a unique block stabilized by μ.

    In what follows, we are going to construct a representative block for each block orbit. A representative block for the block orbit of length 31 will be the block fixed by μ. Hence the multiplicities of orbit numbers in orbit blocks, corresponding to point and block orbit of length 31, will be 0,1 (mod 3).

    The ρfixed block can be written in the form:

    L1=(1011130)(2021230)

    or

    L1=131231.

    Let L2,L3,L4,L5,L6 be the representative blocks for the five non–trivial block orbits. There are exactly two non–fixed orbit blocks passing through the fixed point . Let them be L2,L3. We write

    L2=1a12a23a34a45a5

    L3=1b12b23b34b45b5

    where ai,bi denote the multiplicities of the appearance of orbit numbers in the orbit blocks L2,L3, respectively.

    The multiplicities of the appearance of orbit numbers satisfy the following conditions:

    a1+a2+a3+a4+a5=62,
    b1+b2+b3+b4+b5=62.

    Because |LiL1|=14,i=2,3 and Li,i=1,2,3 we have a1+a2=13, b1+b2=13, and consequently a3+a4+a5=49,b3+b4+b5=49. From (7) we have

    [L2,L2]=31/111+31/31a21+31/31a22+31/31a23+31/93a24+31/93a25
    =1431+6314=483
    [L3,L3]=31/111+31/31b21+31/31b22+31/31b23+31/93b24+31/93b25
    =1431+6314=483
    [L3,L2]=3111+31/31a1b1+31/31a2b2+31/31a3b3+31/93a4b4+31/93a5b5
    =1431=434

    where 0ai13,i=1,2,0a321,0ai38,i=4,5.

    In order to reduce isomorphic cases that may appear in the orbit structures at the last stage, without loss of generality, for block L2, we can use the reduction a1a2,a4a5.

    Using the computer we have proved that there exist only six different orbit types for the block L2 satisfying the above mentioned conditions:

    a1a2a3a4a51.103721212.941021183.94424214.761024155.76727156.7642718

    Further on, acting with the Frobenius group G=F313, for orbit block L3 we have:

    The fourth orbit block L4 has the form:

    L4=1c12c23c34c45c5

    where ci,i=1,2,,5 are multiplicities of the appearance of orbit numbers 1, 2, 3, 4 and 5 in orbit block L4.

    We have: c1+c2+c3+c4+c5=63,

    [L4,L4]=c21+c22+c23+1/3c24+1/3c25=1431+6314=483,
    [L4,Li]=1431=434,(i=2,3).

    [L4L1]=14 implies c1+c2=14, therefore c3+c4+c5=6314=49.

    [L4,L4]=483 implies 0c321, and 0ci38,i=4,5, whereas c1+c2=14 implies 0ci14,i=1,2.

    Further on, acting with the Frobenius group G=F313, for the orbit block L4, for the number of triples L1,L2,L3 given in Table 1, we have:

    Table 1.  Triples {L1,L2,L3}.
    Block L2 Number of orbit types for L3 Number of triples {L1,L2,L3}
    Type 1. 1 1
    Type 2. 1 1
    Type 3. 1 1
    Type 4. 1 1
    Type 5. 1 1
    Type 6. 1 1

     | Show Table
    DownLoad: CSV

    Therefore, we have found twelve compatible triples L2,L3,L4, respectively twelve compatibile quadruples L1,L2,L3,L4.

    Let L5=1d12d23d34d45d5 be the fifth orbit block, where di,i=1,2,,5 denote the multiplicities of the appearance of orbit numbers in the block L5.

    We have:

    d1+d2+d3+d4+d5=63,
    [L5,L5]=93/31d21+93/31d22+93/31d23+93/93d24+93/93d25=1493+6314=1351,
    [L5,Li]=1493=1302,(i=2,3,4).

    |L5L1|=14 implies d1+d2=14, therefore d3+d4+d5=49.

    [L5,L5]=1351 implies 0d321, and 0di36,i=4,5, whereas d1+d2=14 implies 0di14,i=1,2.

    Further on, acting with the Frobenius group G=F313, for the orbit block L5, for the number of triples L2,L3,L4 given in Table 2, we have:

    Table 2.  Triples {L2,L3,L4}.
    Block L2 Number of doubles {L2,L3} Number of triples {L2,L3,L4}
    Type 1. 1 2
    Type 2. 1 2
    Type 3. 1 2
    Type 4. 1 2
    Type 5. 1 2
    Type 6. 1 2

     | Show Table
    DownLoad: CSV

    Obviously, among blocks L5 are also blocks L6. Because of that, we choose doubles among candidates for the block L5, such that every couple of them satisfies the intersection in 14 points. Based on this fact we have found that, from Case 7. and Case 10. in Table 3. for the number of orbit types for L5, up to isomorphism, there are exactly two orbit structures:

    Table 3.  Orbit types for L5.
    Quadruples {L1,L2,L3,L4} Number of orbit types for L5
    Case 1. 1
    Case 2. 1
    Case 3. 1
    Case 4. 1
    Case 5. 1
    Case 6. 1
    Case 7. 2
    Case 8. 1
    Case 9. 1
    Case 10. 2
    Case 11. 1
    Case 12. 1

     | Show Table
    DownLoad: CSV
    Table 4.  Orbit structures.
    OS1. 1 31 31 31 93 93
    1 31 31 0 0 0
    1 7 6 10 24 15
    1 6 7 4 18 27
    0 10 4 10 15 24
    0 8 6 5 24 20
    0 5 9 8 20 21
    OS2. 1 31 31 31 93 93
    1 31 31 0 0 0
    1 7 6 7 27 15
    1 6 7 7 15 27
    0 7 7 1 24 24
    0 9 5 8 20 21
    0 5 9 8 21 20

     | Show Table
    DownLoad: CSV

    Lemma 3.5. Up to isomorphism there are exactly eight orbit structures for symmetric (280, 63, 14) designs and the automorphism group F313 acting with the orbit distribution O3=[1;31;31;31;31;31;31; 93].

    Proof. We put PI={I0,I1,,I30},I=1,2,3,4,5,6, P7={70,71,,730,80,81,,830,90,91,, 930}, for the non–trivial orbits of the group G. Thus, G acts on these point orbits as a permutation group in a unique way. Hence, for the two generators of G we may put

    ρ=()(I0,I1,,I30),I=1,2,,9,

    where is the fixed point of collineation, whereas non–trivial ρorbits are numbers 1,2,3,4,5,6,7, 8,9 and ,10,11,,930 all points of the symmetric block design D, and the collineation μ of order 3 acts in the symmetric block design as permutation (1)(2)(3)(4)(5)(6)(7,8,9) on orbit numbers, whereas on indices acts μ:x5x (mod 31) or

    μ=()(K0)(K1,K5,K25)(K2,K10,K19)(K3,K15,K13)(K4,K20,K7)
    (K6,K30,K26)(K8,K9,K14)(K11,K24,K27)(K12,K29,K21)(K16,K18,K28)
    (K17,K23,K22)(7i,85i,925i),K=1,2,3,4,5,6,i=0,,30.

    We immediately obtain the following.

    Corollary 2. The element μ of G of order 3 fixes precisely 7 points and 7 blocks of D. Each block orbit contains a unique block stabilized by μ.

    In what follows, we are going to construct a representative block for each block orbit. A representative block for the block orbit of length 31 will be the block fixed by μ. Hence the multiplicities of orbit numbers in orbit blocks, corresponding to point and block orbit of length 31, will be 0,1 (mod 3).

    The ρfixed block can be writen in the form:

    L1=(1011130)(2021230)

    or

    L1=131231.

    Let L2,L3,L4,L5,L6,L7,L8 be the representative blocks for the seven non–trivial block orbits. There are exactly two non–fixed orbit blocks passing through the fixed point . Let them be L2,L3. We write

    L2=1a12a23a34a45a56a67a7
    L3=1b12b23b34b45b56b67b7

    where ai,bi denote the multiplicities of the appearance of orbit numbers in the orbit blocks L2,L3, respectively.

    The multiplicities of the appearance of orbit numbers satisfy the following conditions:

    a1+a2+a3+a4+a5+a6+a7=62,
    b1+b2+b3+b4+b5+b6+b7=62.

    Because |LiL1|=14,i=2,3 and Li,i=1,2,3 we have a1+a2=13, b1+b2=13, and consequently a3+a4+a5+a6+a7=49,b3+b4+b5+b6+b7=49. From (7) we have

    [L2,L2]=31/111+31/31a21+31/31a22+31/31a23+31/31a24+31/31a25
    +31/31a26+31/93a27=1431+6314=483
    [L3,L3]=31/111+31/31b21+31/31b22+31/31b23+31/31b24+31/31b25
    +31/31b26+31/93b27=1431+6314=483
    [L3,L2]=3111+31/31a1b1+31/31a2b2+31/31a3b3+31/31a4b4+31/31a5b5
    +31/31a6b6+31/93a7b7=1431=434

    where 0ai13,i=1,2,0ai21,i=3,4,5,6,0a738.

    In order to reduce isomorphic cases that may appear in the orbit structures at the last stage, without loss of generality, for block L2, we can use the reduction a1a2,a3a4a5a6.

    Using the computer we have proved that there exist only ten different orbit types for the block L2 satisfying the above mentioned conditions:

    a1a2a3a4a5a6a71.1037777212.9410777183.9410666214.949976185.947774246.769997157.769994188.769973219.7697632410.76666427

    Further on, acting with the Frobenius group G=F313, for orbit block L3 we have:

    The fourth orbit block L4 has the form:

    L4=1c12c23c34c45c56c67c7

    where ci,i=1,2,,7 are multiplicities of the appearance of orbit numbers 1, 2, 3, 4, 5, 6 and 7 in orbit block L4.

    We have: c1+c2+c3+c4+c5+c6+c7=63,

    [L4,L4]=c21+c22+c23+c24+c25+c26+1/3c27=1431+6314=483,
    [L4,Li]=1431=434,(i=2,3).

    [L4L1]=14 implies c1+c2=14, therefore c3+c4+c5+c6+c7=6314=49.

    [L4,L4]=483 implies 0ci21,i=3,4,5,6, and 0c738, whereas c1+c2=14 implies 0ci14,i=1,2.

    Further on, acting with the Frobenius group G=F313, for the orbit block L4, for the number of triples L1,L2,L3 given in Table 5, we have:

    Table 5.  Triples {L1,L2,L3}.
    Block L2 Number of orbit types for L3 Number of triples {L1,L2,L3}
    Type 1. 1 1
    Type 2. 1 1
    Type 3. 0 0
    Type 4. 0 0
    Type 5. 1 1
    Type 6. 0 0
    Type 7. 0 0
    Type 8. 0 0
    Type 9. 0 0
    Type 10. 0 0

     | Show Table
    DownLoad: CSV

    Note that in set of possible candidates for the orbit block L4 are also orbit blocks L5,L6,L7, because they satisfy the same conditions. Therefore, we investigate quadruples of blocks {L4,L5,L6,L7} which are pairwise compatible. In this way, by computer, for all three cases for the number of orbit types for L4 given in Table 6, we find quadruples {L4,L5,L6,L7}, respectively septuples {L1,L2,L3,L4,L5,L6,L7} and have:

    Table 6.  Quadruples {L1,L2,L3,L4}.
    Triple L1,L2,L3 Number of orbit types for L4 Number of quadruples {L1,L2,L3,L4}
    Case 1. (Type 1 for L2) 100 100
    Case 2. (Type 2 for L2) 28 28
    Case 3. (Type 5 for L2) 28 28

     | Show Table
    DownLoad: CSV

    The eighth orbit block L8 has the form:

    L8=1d12d23d34d45d56d67d7

    where di,i=1,2,,7 are multiplicity of the appearance of orbit numbers 1, 2, 3, 4, 5, 6 and 7 in orbit block L8.

    We have: d1+d2+d3+d4+d5+d6+d7=63,

    [L8,L8]=3d21+3d22+3d23+3d24+3d25+3d26+d27=1493+6314=1351,
    [L8,Li]=1493=1302,(i=2,3,4,5,6,7).

    [L8L1]=14 implies d1+d2=14, therefore d3+d4+d5+d6+d7=6314=49.

    [L8,L8]=1351 implies 0di21,i=3,4,5,6, and 0d736, whereas d1+d2=14 implies 0di14,i=1,2.

    Further on, acting with the Frobenius group G=F313, for the number of septuples {L1,L2,L3,L4, L5,L6,L7} given in Table 7. we find orbit block L8. By computer we found, up to isomorphism, exactly eight orbit structure:

    Table 7.  Septuples {L1,L2,L3,L4,L5,L6,L7}.
    Triple L1,L2,L3 Number of septuples {L1,L2,L3,L4,L5,L6,L7}
    Case 1. 57
    Case 2. 15
    Case 3. 15

     | Show Table
    DownLoad: CSV
    Table 8.  Orbit structures.
    OS3. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 10 3 7 7 7 7 21
    1 3 10 7 7 7 7 21
    0 7 7 13 6 6 6 18
    0 7 7 6 13 6 6 18
    0 7 7 6 6 13 6 18
    0 7 7 6 6 6 13 18
    0 7 7 6 6 6 6 25
    OS4. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 10 3 7 7 7 7 21
    1 3 10 7 7 7 7 21
    0 7 7 13 6 6 6 18
    0 7 7 6 12 9 7 15
    0 7 7 6 9 4 3 27
    0 7 7 6 7 3 12 21
    0 7 7 6 5 9 7 22
    OS5. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 10 3 7 7 7 7 21
    1 3 10 7 7 7 7 21
    0 7 7 13 6 6 6 18
    0 7 7 6 12 9 4 18
    0 7 7 6 9 4 12 18
    0 7 7 6 4 12 9 18
    0 7 7 6 6 6 6 25
    OS6. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 10 3 7 7 7 7 21
    1 3 10 7 7 7 7 21
    0 7 7 12 9 7 6 15
    0 7 7 9 3 6 4 27
    0 7 7 7 6 3 12 21
    0 7 7 6 4 12 9 18
    0 7 7 5 9 7 6 22
    OS7. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 9 4 10 7 7 7 18
    1 4 9 4 7 7 7 24
    0 10 4 4 6 6 6 27
    0 7 7 6 13 6 6 18
    0 7 7 6 6 13 6 18
    0 7 7 6 6 6 13 18
    0 6 8 9 6 6 6 22
    OS8. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 9 4 10 7 7 7 18
    1 4 9 4 7 7 7 24
    0 10 4 4 6 6 6 27
    0 7 7 6 12 9 4 18
    0 7 7 6 9 4 12 18
    0 7 7 6 4 12 9 18
    0 6 8 9 6 6 6 22
    OS9. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 9 4 10 7 7 7 18
    1 4 9 4 7 7 7 24
    0 10 4 3 9 7 6 24
    0 7 7 9 4 3 6 27
    0 7 7 7 3 12 6 21
    0 7 7 6 6 6 13 18
    0 6 8 8 9 7 6 19
    OS10. 1 31 31 31 31 31 31 93
    1 31 31 0 0 0 0 0
    1 9 4 10 7 7 7 18
    1 4 9 4 7 7 7 24
    0 10 4 3 9 7 6 24
    0 7 7 9 3 6 4 27
    0 7 7 7 6 3 12 21
    0 7 7 6 4 12 9 18
    0 6 8 8 9 7 6 19

     | Show Table
    DownLoad: CSV

    Lemma 3.6. Up to isomorphism there are exactly three orbit structures for symmetric (280, 63, 14) designs and the automorphism group F313 acting with the orbit distribution O4=[1;31;31;31;31;31;31; 31,31,31].

    Proof. We put PI={I0,I1,,I30},I=1,2,3,4,5,6,7,8,9, for the non–trivial orbits of the group G. Thus, G acts on these point orbits as a permutation group in a unique way. Hence, for the two generators of G we may put

    ρ=()(I0,I1,,I30),I=1,2,,9,

    where is the fixed point of collineation, whereas non–trivial ρorbits are numbers 1,2,3,4,5,6,7, 8,9 and ,10,11,,930 all points of the symmetric block design D, and the collineation μ of order 3 acts in the symmetric block design as permutation (1)(2)(3)(4)(5)(6)(7)(8)(9) on orbit numbers, whereas on indices acts μ:x5x (mod 31) or

    μ=()(K0)(K1,K5,K25)(K2,K10,K19)(K3,K15,K13)(K4,K20,K7)
    (K6,K30,K26)(K8,K9,K14)(K11,K24,K27)(K12,K29,K21)
    (K16,K18,K28)(K17,K23,K22),K=1,2,3,4,5,6,7,8,9.

    We immediately obtain the following.

    Corollary 3. The element μ of G of order 3 fixes precisely 10 points and 10 blocks of D. Each block orbit contains a unique block stabilized by μ.

    In what follows, we are going to construct a representative block for each block orbit. A representative block for the block orbit of length 31 will be the block fixed by μ. Hence the multiplicities of orbit numbers in orbit blocks, will be 0,1 (mod 3).

    The ρfixed block can be writen in the form:

    L1=(1011130)(2021230)

    or

    L1=131231.

    Let L2,L3,L4,L5,L6,L7,L8,L9,L10 be the representative blocks for the nine non–trivial block orbits. There are exactly two non–fixed orbit blocks passing through the fixed point . Let them be L2,L3. We write

    L2=1a12a23a34a45a56a67a78a89a9
    L3=1b12b23b34b45b56b67b78b89b9

    where ai,bi denote the multiplicities of the appearance of orbit numbers in the orbit blocks L2,L3, respectively.

    The multiplicities of the appearances of orbit numbers satisfy the following conditions:

    a1+a2+a3+a4+a5+a6+a7+a8+a9=62.
    b1+b2+b3+b4+b5+b6+b7+b8+b9=62.

    Because |LiL1|=14,i=2,3 and Li,i=1,2,3 we have a1+a2=13, b1+b2=13, and consequently a3+a4+a5+a6+a7+a8+a9=49,b3+b4+b5+b6+b7+b8+b9=49. From (7) we have

    [L2,L2]=31/111+31/31a21+31/31a22+31/31a23+31/31a24+31/31a25+31/31a26
    +31/31a27+31/31a28+31/31a29=1431+6314=483
    [L3,L3]=31/111+31/31b21+31/31b22+31/31b23+31/31b24+31/31b25+31/31b26
    +31/31b27+31/31b28+31/31b29=1431+6314=483
    [L3,L2]=3111+31/31a1b1+31/31a2b2+31/31a3b3+31/31a4b4+31/31a5b5
    +31/31a6b6+31/31a7b7+31/31a8b8+31/31a9b9=1431=434

    where 0ai13,i=1,2,0ai21,i=3,4,,9.

    In order to reduce isomorphic cases that may appear in the orbit structures at the last stage, without loss of generality, for block L2, we can use the reduction a1a2,a3a4a5a6a7a8a9.

    Using the computer we have proved that there exist only six different orbit types for the block L2 satisfying the above mentioned conditions:

    a1a2a3a4a5a6a7a8a91.10377777772.94107776663.9499766664.76109776645.7699966646.769977773

    Further on, acting with the Frobenius group G=F313, for orbit block L3 we have:

    Hence, we have only one double L2,L3, respectively only one triple L1,L2,L3:

    1313131313131313131L1131310000000L211037777777L313107777777

    The fourth orbit block L4 has the form:

    L4=1c12c23c34c45c56c67c78c89c9

    where ci,i=1,2,,9 are multiplicities of the appearance of orbit numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 in orbit block L4.

    We have: c1+c2+c3+c4+c5+c6+c7+c8+c9=63,

    [L4,L4]=c21+c22+c23+c24+c25+c26+c27+c28+c29=1431+6314=483,
    [L4,Li]=1431=434,(i=2,3).

    [L4L1]=14 implies c1+c2=14, therefore c3+c4+c5+c6+c7+c8+c9=6314=49.

    [L4,L4]=483 implies 0ci21,i=3,4,5,6,7,8,9, whereas c1+c2=14 implies 0ci14,i=1,2.

    Using the computer we have proved that for the number of triples L1,L2,L3 given in Table 9., there exist exactly 2527 dfferent orbit types for the block L4 satisfying the above mentioned conditions:

    c1c2c3c4c5c6c7c8c91.77136666662.77129666643.77129666462525.77347791092526.77347799102527.773469999
    Table 9.  Triples {L1,L2,L3}.
    Block L2 Number of orbit types for L3 Number of triples {L1,L2,L3}
    Type 1. 1 1
    Type 2. 0 0
    Type 3. 0 0
    Type 4. 0 0
    Type 5. 0 0
    Type 6. 0 0

     | Show Table
    DownLoad: CSV

    Note that in set of possible candidates for the orbit block L4 are also orbit blocks L5,L6,L7,L8 and L9, because they satisfy the same conditions. Therefore, we investigate sextuples of blocks {L4,L5,L6,L7,L8,L9} which are pairwise compatible. In this way, by computer, we find sextuples {L4,L5,L6,L7,L8,L9}, respectively all orbit structures. Thus, up to isomorphism, we have exactly three orbit structure:

    Thus we have

    Theorem 3.7. Up to isomorphism, there are exactly thirteen orbit structures for a symmetric block design with parameters (280,63,14) admitting the Frobenius Group G=ρ,μ|ρ31=μ3=1,ρμ=ρ5 of order 93; two with the orbit distribution [1;31;31;31;93;93] (Table 4.), eight with the orbit distribution [1;31;31;31;31;31;31;93] (Table 8.) and three with the orbit distribution [1;31;31;31;31;31;31;31;31;31] (Table 10.).

    Table 10.  Orbit structures.
    OS11. 1 31 31 31 31 31 31 31 31 31
    1 31 31 0 0 0 0 0 0 0
    1 10 3 7 7 7 7 7 7 7
    1 3 10 7 7 7 7 7 7 7
    0 7 7 13 6 6 6 6 6 6
    0 7 7 6 13 6 6 6 6 6
    0 7 7 6 6 13 6 6 6 6
    0 7 7 6 6 6 13 6 6 6
    0 7 7 6 6 6 6 13 6 6
    0 7 7 6 6 6 6 6 13 6
    0 7 7 6 6 6 6 6 6 13
    OS12. 1 31 31 31 31 31 31 31 31 31
    1 31 31 0 0 0 0 0 0 0
    1 10 3 7 7 7 7 7 7 7
    1 3 10 7 7 7 7 7 7 7
    0 7 7 13 6 6 6 6 6 6
    0 7 7 6 13 6 6 6 6 6
    0 7 7 6 6 13 6 6 6 6
    0 7 7 6 6 6 13 6 6 6
    0 7 7 6 6 6 6 12 9 4
    0 7 7 6 6 6 6 9 4 12
    0 7 7 6 6 6 6 4 12 9
    OS13. 1 31 31 31 31 31 31 31 31 31
    1 31 31 0 0 0 0 0 0 0
    1 10 3 7 7 7 7 7 7 7
    1 3 10 7 7 7 7 7 7 7
    0 7 7 13 6 6 6 6 6 6
    0 7 7 6 12 9 6 6 6 4
    0 7 7 6 9 4 6 6 6 12
    0 7 7 6 6 6 12 9 4 6
    0 7 7 6 6 6 9 4 12 6
    0 7 7 6 6 6 4 12 9 6
    0 7 7 6 4 12 6 6 6 9

     | Show Table
    DownLoad: CSV

    Remark 1. The actual indexing of these thirteen orbit structures in order to produce an example is still an open problem.

    The author would like to thank the referees for their suggestions and comments, which have improved the overall presentation.

    The author declares there is no conflicts of interest in this paper.



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  • This article has been cited by:

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