EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 3, 2018, 645-651
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On the symmetric block design with parameters
(306,61,12) admitting a group of order 61

Menderes Gashi

Department of Mathematics, Faculty of Mathematics and Natural Sciences,
University of Prishtina, Avenue Mother Teresa 5, 10000 Prishtina, Kosova

Abstract. In this paper we have proved that up to isomorphism there are exactly two orbit
structures for a putative symmetric block design D with parameters (306,61,12), constructed by
group G of order 61. Also the full automorphism groups for these orbit structures are given.

2010 Mathematics Subject Classifications: 05B05

Key Words and Phrases: Symmetric block design, Orbit structure, Automorphism group

1. Introduction and Preliminaries

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” treat-
ments even for a single parameter set (v, k, λ). The most famous instance of this circum-
stance is perhaps the non-existence of the projective plane of order 10, see [10].

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 [3], [5]).

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.

DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3259

Email address: menderes gashi@yahoo.com (M. Gashi)

http://www.ejpam.com 645 c© 2018 EJPAM All rights reserved.



M. Gashi / Eur. J. Pure Appl. Math, 11 (3) (2018), 645-651 646

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 [7] ; see also [6, 8].

The present paper is concerned with a symmetric design D = (P,B, I) with parameters
(306, 61, 12): 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 61. We assume the reader is familiar with the basic facts of design theory,
see for instance [9], [2] and [11]. 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 [11, 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 [11, Corollary 3.7, p. 82]:

|F (g)| ≤ k +
√
k − λ. (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: [11, Theorem 3.3, p.79].
Denote that number by t.

We adopt the notation and terminology of Section 1 in [4]: we repeat some fundamental
relations here for the reader’s sake. Let D be a symmetric design with parameters (v, k, λ)
and let G be a subgroup of the automorphism group AutD 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,
t∑

r=1

ωr =
t∑

i=1

Ωi = 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,

t∑
r=1

γir = k and
t∑

j=1

Γjs = k. (3)

By [2, 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 [2, p.210]. Thus, the
following equations hold:

Ωi · γir = ωr · Γir, (4)

t∑
r=1

γirΓjr = λΩj + δij(k − λ), (5)

t∑
i=1

Γirγis = λωs + δrs(k − λ), (6)



M. Gashi / Eur. J. Pure Appl. Math, 11 (3) (2018), 645-651 647

where δij , δrs are the Kronecker symbols.
For a proof of these equations, the reader is referred to [2] and [4]. Equation (5),

together with (4) yields
t∑

r=1

Ωj

ωr
γirγjr = λΩj + δij(k − λ). (7)

Definition 1. The (t× t)-matrix (γir) is called the orbit structure of the design D.

An automorphism of a orbit structure is a permutation of rows followed by a per-
mutation of columns leaving that matrix unchanged. It is clear that the set of all such
automorphisms is a group, which we call the automorphism group of that orbit structure.

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.

2. Main results

Denote D the symmetric block design with parameters (306,61,12). Since v = 1+5 ·61,
in order to construct the symmetric block design D we use the the cyclic group G =
〈ρ|ρ61 = 1〉 of order 61 as a collineation group.

Lemma 1. Let ρ be an element of G with o(ρ) = 61. Then 〈ρ〉 fixes precisely one point
and one block.

Proof. By [11, Theorem 3.1] the group 〈ρ〉 fixes the same number of points and blocks.
Denote that number by f. Obviouslyf ≡ 306(mod 61), i.e.f ≡ 1(mod 61). The upper
bound (1) for the number of fixed points yeilds f ∈ {1, 62}. 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 = 62 and λ = 12 (there is no k ∈ IN which
satisfies 12 · (v − 1) = k · (k − 1)). Hence, f is equal to 1.

We put PI = {I0, I1, · · · , I60}, I = 1, 2, 3, 4, 5, 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 generator of G we may put

ρ = (∞)(I0, I1, · · · , I60), I = 1, 2, 3, 4, 5,

where ∞ is the fixed point of collineation, whereas non–trivial 〈ρ〉-orbits are numbers 1,
2, 3, 4, 5 and ∞, 10, 11, · · · , 560 are all points of the symmetric block design D.

In what follows, we are going to construct a representative block for each block orbit.
The 〈ρ〉−fixed block can be writen in the form:

L1 = (1011 · · · 160)



M. Gashi / Eur. J. Pure Appl. Math, 11 (3) (2018), 645-651 648

or
L1 = 161.

Let L2, L3, L4, L5, L6 be the representative blocks for the five non–trivial block orbits.
The second orbit block L2 of design D, constructed by collineation can be written as

L2 =∞1a12a23a34a45a5 ,

where ai, i = 1, 2, 3, 4, 5 denote the multiplicities of the appearance of orbit numbers
1, 2, 3, 4 and 5 in the orbit block L2.

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

a1 + a2 + a3 + a4 + a5 = 60,

Because |L1 ∩ L2| = 12, we have a1 = 12. From (7) we have
[L2, L2] = 61/1 · 1 · 1 + 61/61 · a21 + 61/61 · a22 + 61/61 · a23 + 61/61 · a24 + 61/61 · a25

= 12 · 61 + 61− 12 = 781, i.e.

a21 + a22 + a23 + a24 + a25 = 781

or
a22 + a23 + a24 + a25 = 576.

From the last relation, for the multiplicities of appearance in the block L2, we obtain
the reductions 0 ≤ ai ≤ 24, i = 2, 3, 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 may assume that the inequalty
a2 ≥ a3 ≥ a4 ≥ a5 hold.

Using the computer we have proved that there exists exactely one orbit type for the
block L2 that satisfies the above mentioned conditions:

a1 a2 a3 a4 a5
1. 12 12 12 12 12

The third orbit block L3, constructed with the collineation ρ, has the form:

L3 = 1b12b23b34b45b5 ,

where bi, i = 1, 2, · · · , 5 are multiplicities of the appearance of orbit numbers 1,2,3, 4 and
5 in orbit block L3.

The multiplicities of orbit numbers satisfy the following conditions: b1 + b2 + b3 + b4 +
b5 = 61.

[L1 ∩ L3] = 12 implies b1 = 12. From (7) we hawe
[L3, L3] = b21 + b22 + b23 + b24 + b25 = 12 · 61 + 61− 12 = 781
or
b22 + b23 + b24 + b25 = 637.
From the last relation we obtain the reductions 0 ≤ bi ≤ 25, i = 2, 3, 4, 5.



M. Gashi / Eur. J. Pure Appl. Math, 11 (3) (2018), 645-651 649

[L2, L3] = a1b1 + a1b1 + a1b1 + a1b1 + a1b1 = 12 · 61 = 732.
Using the computer we have proved that there are exactly twenty–eight orbit types for

the block L3 satisfying the above mentioned conditions:

b1 b2 b3 b4 b5
1. 12 16 14 11 8
2. 12 16 14 8 11
3. 12 16 11 14 8
4. 12 16 11 8 14
5. 12 16 8 14 11
6. 12 16 8 11 14
7. 12 14 16 11 8
8. 12 14 16 8 11
9. 12 14 14 14 7

10. 12 14 14 7 14
11. 12 14 11 16 8
12. 12 14 11 8 16
13. 12 14 8 16 11
14. 12 14 8 11 16
15. 1 214 7 14 14
16. 12 11 16 14 8
17. 12 11 16 8 14
18. 12 11 14 16 8
19. 12 11 14 8 16
20. 12 11 8 16 14
21. 12 11 8 14 16
22. 12 8 16 14 11
23. 12 8 16 11 14
24. 12 8 14 16 11
25. 12 8 14 11 16
26. 12 8 11 16 14
27. 12 8 11 14 16
28. 12 7 14 14 14

It is clear that among the candidates for the block L3 are also blocks L4, L5, L6. There-
fore, we investigate quadruples of blocks {L3, l4, L5, L6} which are pairwise compatible. In
this way, we have found that, up to isomorphism, there are exactely two orbit structures
for the symmetric block design with parameters (306, 61, 12) acting with the collineation
ρ of order 61:

First orbit structure:



REFERENCES 650

SO1 1 61 61 61 61 61

0 61 0 0 0 0
1 12 12 12 12 12
0 12 16 14 11 8
0 12 14 7 14 14
0 12 11 14 8 16
0 12 8 14 16 11

Directly from orbit structure we find these automorphisms:
1. (1)(L1)
2. (3 5 6)(L3 L6 L5)
3. (3 6 5)(L3 L5 L6)

and the full automporphism group of the orbit stucture SO1 is:

Aut(SO1) = {1, (3 5 6)(3̄ 6̄ 5̄), (3 6 5)(3̄ 5̄ 6̄)}.

Second orbit structure:

SO2 1 61 61 61 61 61

0 61 0 0 0 0
1 12 12 12 12 12
0 12 14 14 14 7
0 12 14 14 7 14
0 12 14 7 14 14
0 12 7 14 14 14

Full automporphism group of the orbit stucture SO2 is:
Aut(SO2) ∼= Σ{3,4,5,6} of order |Aut(SO2)| = 24.
Thus we have

Theorem 1. Up to isomorphism, there are exactly two orbit structures for the symmetric
block design D with parameters (306, 61, 12) admitting the group G of order 61.

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

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