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\begin{document}

\begin{titlepage} 

\vspace*{3.0cm}

{\par\centering \textbf{\large Unique symmetric (66,26,10) design admitting}\\
 \textbf{\large an automorphism of order 55}{\large \par}}{\large \par}

\vspace*{2.0cm}

{\par\centering Dean Crnkovi\'{c}\\
 Department of mathematics \\
 Faculty of philosophy in Rijeka\\
 Omladinska 14, 51000 Rijeka, Croatia\\
 (e-mail: deanc@mapef.pefri.hr) \par}

\smallskip

{\par\centering and \par}

\smallskip

{\par\centering Sanja Rukavina \\
 Department of mathematics \\
 Faculty of philosophy in Rijeka\\
 Omladinska 14, 51000 Rijeka, Croatia\\
 (e-mail: sanjar@mapef.pefri.hr) \par}

\vspace*{2.0cm}

\textit{Running head\/:} \( Z_{55} \) acting on 2-(66,26,10) design

\newpage

\begin{abstract}
We have proved that the first known symmetric (66,26,10) design, constructed by Tran van Trung, 
is up to isomorphism the only symmetric (66,26,10) design admitting an automorphism
of order 55. Full automorphism group of that design is isomorphic to 
\( Frob_{55}\times D_{10} \). 
\end{abstract}
\vspace*{1.5cm}

\textit{Keywords\/:} symmetric design, automorphism group, orbit structure.\\


\textit{Mathematical subject classification} (1991): 05B05.

\end{titlepage}


\section{Introduction and preliminaries}

A symmetric \( (v,k,\lambda ) \) design is a finite incidence structure \( ({\mathcal{P},\mathcal{B}},I) \),
where \( {\mathcal{P}} \) and \( {\mathcal{B}} \) are disjoint sets and \( I\subseteq {\mathcal{P}}\times {\mathcal{B}} \),
with the following properties:

\begin{description}
\item [1.]\( |{\mathcal{P}}|=|{\mathcal{B}}|=v \), 
\item [2.]every element of \( {\mathcal{B}} \) is incident with exactly \( k \)
elements of \( {\mathcal{P}} \), 
\item [3.]every pair of elements of \( {\mathcal{P}} \) is incident with exactly
\( \lambda  \) elements of \( {\mathcal{B}} \). 
\end{description}
Elements of the set \( {\mathcal{P}} \) are called points and elements of the set \( {\mathcal{B}} \) are called blocks.

Let \( {\mathcal{D}=(\mathcal{P},\mathcal{B}},I) \) be a symmetric \( (v,k,\lambda ) \)
design and \( G\leq Aut{\mathcal{D}} \). Group \( G \) has the same number
of point and block orbits. Let us denote the number of \( G- \)orbits by \( t \),
point orbits by \( {\mathcal{P}}_{1},\ldots ,{\mathcal{P}}_{t} \), block orbits
by \( {\mathcal{B}}_{1},\ldots ,{\mathcal{B}}_{t} \), and put \( |{\mathcal{P}}_{r}|=\omega _{r} \),
\( |{\mathcal{B}}_{i}|=\Omega _{i} \). We shall denote points of the orbit
\( {\mathcal{P}}_{r} \) by \( r_{0},\ldots ,r_{\omega _{r}-1} \), (i.e. \( {\mathcal{P}}_{r}=\{r_{0},\ldots ,r_{\omega _{r}-1}\} \)).
Further, denote by \( \gamma _{ir} \) the number of points of \( {\mathcal{P}}_{r} \)
which are incident with the representative of the block orbit \( {\mathcal{B}}_{i} \).
For those numbers the following equalities hold: 
\begin{eqnarray}
\sum _{r=1}^{t}{\gamma }_{ir} & = & k\, ,\\
\sum _{r=1}^{t}\frac{{\Omega }_{j}}{{\omega }_{r}}{\gamma }_{ir}{\gamma }_{jr} & = & \lambda {\Omega }_{j}+{\delta }_{ij}\cdot (k-\lambda )\, .
\end{eqnarray}


\begin{defi}The \((t \times t)\)-matrix \(({\gamma}_{ir})\) with entries satisfying properties \((1)\) and \((2)\) is called the orbit structure for parameters 
\((v,k, \lambda)\) and orbit
distribution \((\omega_{1}, \ldots ,\omega_{t})\), 
\((\Omega_{1}, \ldots ,\Omega_{t})\).
\end{defi}

The first step of the construction of designs is to find all orbit structures
\( (\gamma _{ir}) \) for some parameters and orbit distribution. The next step,
called indexing, is to determine for each number \( \gamma _{ir} \) exactly
which points from the point orbit \( {\mathcal{P}}_{r} \) are incident with
representative of the block orbit \( {\mathcal{B}}_{i} \). Because of the large
number of possibilities, it is often necessary to involve a computer in both
steps of the construction.

\begin{defi}The set of indices of points of the orbit \({\mathcal{P}}_{r}\) indicating which points of \({\mathcal{P}}_{r}\) are incident with the 
representative of the block orbit \({\mathcal{B}}_{i}\) is called the index set for the position \((i,r)\) of the orbit structure.
\end{defi}

First symmetric \( (66,26,10) \) design is constructed by Tran van Trung (see
{[}8{]}). Full automorphism group of that design is isomorphic to $ Frob_{55}\times D_{10} $. 
I. Matuli\'c-Bedeni\'c, K. Horvati\'c-Baldasar and E. Kramer (see {[}6{]}) have constructed 18 mutually nonisomorphic symmetric \( (66,26,10) \) 
designs with Frobenius automorphism group of order 39. Later on, M.-O. Pav\v cevi\'c and E. Spence (see {[}7{]}) have proved that there are 
up to isomorphism 558 symmetric \( (66,26,10) \) designs admitting a dihedral 
automorphism group of order 10 acting with orbit distribution $(1,5,5,5,5,5,5,5,5,5,5)$ 
on the sets of points and blocks, and 22 symmetric \( (66,26,10) \) designs having an elementary
abelian automorphism group of order 25. Finally, 
it has been proved (see {[}2{]}) that  there are up to isomorphism three symmetric \( (66,26,10) \) designs with automorphism
group isomorphic to \( Frob_{55} \).  Among constructed symmetric \( (66,26,10) \) designs there are 590 mutually nonisomorphic ones.

Our aim is to construct symmetric \( (66,26,10) \) designs admitting an abelian automorphism group of order 55, which will complete classification
of symmetric \( (66,26,10) \) designs with an automorphism group of order 55.


\section{\protect\protect\protect\( Z_{55}\protect \protect \protect \) acting on
a symmetric (66,26,10) design}

From now on we shall denote by \( G \) a cyclic group of order 55  presented as follows:
\[
G\cong \langle \rho ,\sigma \mid {\rho }^{11}=1,\, {\sigma }^{5}=1,\, {\rho }^{\sigma }=1\rangle .\]


Let \( \alpha  \) be an automorphism of a symmetric design. We shall denote
by \( F(\alpha ) \) the number of points fixed by \( \alpha  \). In that case,
the number of blocks fixed by \( \alpha  \) is also \( F(\alpha ) \).

\begin{lema}Let \(\rho\) be an automorphism of a symmetric \((66,26,10)\) design. If \(|\rho|=11\), then \(F(\rho)=0\).\end{lema}
\textbf{Proof} \enspace It is known that \( F(\rho )<k+\sqrt{k-\lambda } \)
and \( F(\rho )\equiv v(mod\, \, |\rho |) \). Therefore, \( F(\rho )\in \{0,11,22\} \).
One can not construct fixed blocks for \( F(\rho )\in \{11,22\} \).\( \diamond  \)

\begin{lema}Let \(G\) be a cyclic automorphism group of order \(55\) of a symmetric \((66,26,10)\) design \(\mathcal{D}\). 
 \(G\) acts semistandardly on \(\mathcal{D}\) with orbit distribution  \((11,55)\).
\end{lema} \textbf{Proof} \enspace Group \( \langle \rho \rangle  \)
acts on \( \mathcal{D} \) with orbit distribution \( (11,11,11,11,11,11) \).
Since \( \langle \rho \rangle \triangleleft G \), \( \sigma  \) maps \( \langle \rho \rangle  \)-orbits
on \( \langle \rho \rangle  \)-orbits. Therefore, only possibilities for orbit
distributions for the group $G$ 
are \( (11,11,11,11,11,11) \) and \( (11,55) \).  In the case of orbit distribution  \( (11,11,11,11,11,11) \) automorphism 
\( \sigma  \) would act on ${\cal D}$ with 66 fixed points (blocks), which is not possible. For orbit distribution \( (11,55) \) 
one can construct only one orbit structure, namely:

{\par\centering 

\begin{tabular}{c|cc}
 OS &
 11 &
 55 \\
\hline 
11 &
 1 &
 25 \\
 55 &
 5 &
 21  \\
\end{tabular}

\par}
\( \diamond  \)



\section{Construction of the design}

It would be difficult to proceed with indexing of orbit structure OS. For example,
there are \( {55\choose 25} \) possibilities for index sets for the position
\( (1,2) \) in the OS. Therefore, we shall use a principal series \( \langle 1\rangle \triangleleft \langle \rho \rangle \triangleleft G \)
of the automorphism group \( G \). Our aim is to find all orbit structures
for the group \( \langle \rho \rangle  \) corresponding to the structure OS.
We shall construct designs from those orbit structures for \( \langle \rho \rangle  \),
having in mind the action of the permutation \( \sigma  \) on \( \langle \rho \rangle  \)-orbits.

\begin{tm}Up to isomorphism  there is only one symmetric \((66,26,10)\) design with an automorphism of order $55$. 
Full automorphism group of that design is isomorphic to \(Frob_{55} \times D_{10}\) and its \(2\)-rank is \(31\). 
\end{tm} \textbf{Proof} \enspace The only orbit structure for the group
\( \langle \rho \rangle  \) corresponding to the orbit structure OS is\\

{\par\centering 

\begin{tabular}{c|cccccc}
 OS1 &
 11 &
 11 &
 11 &
 11 &
 11 &
 11 \\
\hline 
11 &
 1 &
 5 &
 5 &
 5 &
 5 &
 5 \\
 11 &
 5 &
 5 &
 5 &
 5 &
 5 &
 1 \\
 11 &
 5 &
 5 &
 5 &
 5 &
 1 &
 5 \\
 11 &
 5 &
 5 &
 5 &
 1 &
 5 &
 5 \\
 11 &
 5 &
 5 &
 1 &
 5 &
 5 &
 5 \\
 11 &
 5 &
 1 &
 5 &
 5 &
 5 &
 5  \\
\end{tabular}

\par}
\bigskip

We shall proceed
with indexing of the orbit structure OS1, knowing that \( \sigma  \) acts on the
set of six \( \langle \rho \rangle  \)-orbits of blocks as a permutation \( (1)(2,3,4,5,6) \),
and on the set of point orbits as \( (1)(2,6,5,4,3) \). It is sufficient to
determine index sets for the first and second row of orbit structure OS1. Also,
in the first row we have to determine index sets only for positions \( (1,1) \)
and \( (1,2) \). We shall assume that automorphisms \( \rho  \) and \( \sigma  \)
act on the set of points as follows:

\begin{description}
\item [\( \rho =(I_{0},I_{1},\ldots ,I_{10}),\, \, I=1,2,\ldots ,6, \)]~
\item [\( \sigma =(1_{i})(2_{i},6_{i},5_{i},4_{i},3_{i}) \),]\( i=0,1,\ldots ,10 \).~
\end{description}
Because of that action, index sets for positions $(1,2)$ and $(2,1)$ have to be basic blocks for
symmetric $(11,5,2)$ designs. 
As representatives of the first and second block orbit we shall choose first blocks in those orbits, 
with respect to the lexicographical order. Index sets which could occur in designs are: 
\[
0=\{0\},\, \, \ldots \, \, ,\, \, 10=\{10\},\, \, 11=\{0,1,2,3,4\},\, \, \ldots \, \, ,\, \, 472=\{6,7,8,9,10\}.\]


To eliminate isomorphic structures during the indexing, we have been using 
automorphisms of the orbit structure OS1 which commute
with permutation representation of the \( \sigma  \) on the set of \( \langle \rho \rangle  \)-orbits,
and permutations \( \alpha_l,\beta _{1,k},\beta _{2,k}\in N_{S}(G),\, \, 2\leq l\leq 10,\ 1\leq k\leq 10 \), 
which act on the set of points as follows:

\begin{description}
\item [\( r_{i}\alpha _{l}=r_{j},j\equiv (i \cdot l)(mod\, \, 11),\, \, for\, \, r=1, \ldots ,6,\ i=0,1,\ldots ,10 \),]~
\item [\( 1_{i}\beta _{1,k}=1_{j},j\equiv (i+k)(mod\, \, 11),\, \, for\, \, i=0,1,\ldots ,10, \)]~
\item [\( r_{i}\beta _{1,k}=r_{i}, \, for\, \, r=2, \ldots ,6,\ i=0,1,\ldots ,10 \),]~
\item [\( 1_{i}\beta _{2,k}=1_{i},\, \, for\, \, i=0,1,\ldots ,10, \)]~
\item [\( r_{i}\beta _{2,k}=r_{j},j\equiv (i+k)(mod\, \, 11),\, \, for\, \, r=2, \ldots ,6,\ i=0,1,\ldots ,10 \),]~
\end{description}
We have constructed up to isomorphism only one design, which is presented in 
terms of index sets as follows:\\

{\centering \begin{tabular}{cccccc}
0 &
 280 &
 280 &
 280 &
 280 &
 280  \\
 20 &
 20 &
 450 &
 450 &
 20 &
 5 \\
 20 &
 450 &
 450 &
 20 &
 5 &
 20 \\
 20 &
 450 &
 20 &
 5 &
 20 &
 450 \\
 20 &
 20 &
 5 &
 20 &
 450 &
 450 \\
 20 &
 5 &
 20 &
 450 &
 450 &
 20 \\
\end{tabular}\par}
\bigskip

The group $Z_{55}$ is subgroup of the group $Frob_{55} \times D_{10}$.   
Therefore, it is clear that constructed design is isomorphic to the one constructed by Tran van Trung with
full automorphism group isomorphic to \(Frob_{55} \times D_{10}\).
With the help of the computer program by V. Tonchev we have computed that \(2\)-rank 
of the design is \(31\). Constructed design is self-dual.
\( \diamond  \)

In [2] we have proved the following theorem:

\begin{tm}
Up to isomorphism  there are three symmetric \((66,26,10)\) designs with automorphism group \(Frob_{55}\). 
Let us denote them by \({\mathcal{D}}_{1}\), \({\mathcal{D}}_{2}\) and \({\mathcal{D}}_{3}\).
Full automorphism groups of designs \({\mathcal{D}}_{1}\) and \({\mathcal{D}}_{2}\) are isomorphic 
to \(Frob_{55}\), and full automorphism group of the design \({\mathcal{D}}_{3}\) is isomorphic to \(Frob_{55} \times D_{10}\).
Designs \({\mathcal{D}}_{1}\) and \({\mathcal{D}}_{2}\) are dually isomorphic. \(2-\)ranks of these three designs are \(31\).
\end{tm} 

\noindent {\bf Remark} \enspace Design \({\mathcal{D}}_{3}\) is isomorphic to the design constructed by Tran van Trung. 

Thereby we have proved the following statement:

\begin{tm}
Up to isomorphism there are three symmetric \((66,26,10)\) designs with an automorphism group of order 55, namely the designs from theorem $2$.
\end{tm} 


\begin{thebibliography}{10}
\bibitem{a}M. Aschbacher, On Collineation Groups of Symmetric Block Designs, J. Combin.
Theory 11 (1971), 272--281. 
\bibitem{cr}D. Crnkovi\'c, S. Rukavina, Symmetric (66,26,10) designs having \( Frob_{55} \) 
as an automorphism group, Glas. Mat., III. Ser. Vol. 35(55) (2000), 271-281.
\bibitem{cy}V. \'{C}epuli\'{c}, On symmetric block designs (40,13,4) with automorphisms
of order 5, Discrete Math. 128 (1994) no. 1--3, 45--60. 
\bibitem{j}Z. Janko, Coset Enumeration in Groups and Constructions of Symmetric Designs,
Combinatorics '90 (1992), 275--277. 
\bibitem{l}E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press
(1983). 
\bibitem{mhk}I. Matuli\'{c}-Bedeni\'{c}, K. Horvati\'{c}-Baldasar, E. Kramer, Construction
of new symmetric designs with parameters (66,26,10), J. Comb. Des. 3 (1995)
405--410. 
\bibitem{ps}M.-O. Pav\v{c}evi\'{c}, E. Spence, Some new symmetric designs with \( \lambda = \)
10 having an automorphism of order 5, Discrete Math. 196 (1999) 257--266. 
\bibitem{t}Tran van Trung, The existence of symmetric block designs with parameters (41,16,6)
and (66,26,10), J. Comb. Theory A 33 (1982) 201--204. 
\end{thebibliography}
\end{document}