May  2011, 5(2): 303-308. doi: 10.3934/amc.2011.5.303

On the non-existence of sharply transitive sets of permutations in certain finite permutation groups

1. 

Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, 97074 Würzburg, Germany

2. 

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary

Received  April 2010 Revised  November 2010 Published  May 2011

In this short note we present a simple combinatorial trick which can be effectively applied to show the non-existence of sharply transitive sets of permutations in certain finite permutation groups.
Citation: Peter Müller, Gábor P. Nagy. On the non-existence of sharply transitive sets of permutations in certain finite permutation groups. Advances in Mathematics of Communications, 2011, 5 (2) : 303-308. doi: 10.3934/amc.2011.5.303
References:
[1]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[2]

A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular graphs, in "Ergebnisse der Mathematik und ihrer Grenzgebiete,'' Springer-Verlag, Berlin, 1989.

[3]

P. Dembowski, "Finite Geometries,'' Springer-Verlag, Berlin-New York, 1968.

[4]

P. Frankl and M. Deza, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A, 22 (1977), 352-360. doi: 10.1016/0097-3165(77)90009-7.

[5]

GAP Group, "GAP - Groups, Algorithms, and Programming,'' University of St Andrews and RWTH Aachen, 2002, Version 4r3.

[6]

T. Grundhöfer, The groups of projectivities of finite projective and affine planes, in "Eleventh British Combinatorial Conference (London, 1987),'' Ars Combin., 25 (1988), 269-275.

[7]

T. Grundhöfer and P. Müller, Sharply 2-transitive sets of permutations and groups of affine projectivities, Beiträge Algebra Geom., 50 (2009), 143-154.

[8]

M. E. O'Nan, Sharply 2-transitive sets of permutations, in "Proc. Rutgers Group Theory Year, 1983-1984 (New Brunswick, N.J., 1983-1984),'' Cambridge Univ. Press, (1985), 63-67.

[9]

J. Quistorff, A survey on packing and covering problems in the Hamming permutation space, Electron. J. Combin., 13 (2006), 13.

[10]

H. Tarnanen, Upper bounds on permutation codes via linear programming, Europ. J. Combinatorics, 20 (1999), 101-114. doi: 10.1006/eujc.1998.0272.

show all references

References:
[1]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[2]

A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-regular graphs, in "Ergebnisse der Mathematik und ihrer Grenzgebiete,'' Springer-Verlag, Berlin, 1989.

[3]

P. Dembowski, "Finite Geometries,'' Springer-Verlag, Berlin-New York, 1968.

[4]

P. Frankl and M. Deza, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A, 22 (1977), 352-360. doi: 10.1016/0097-3165(77)90009-7.

[5]

GAP Group, "GAP - Groups, Algorithms, and Programming,'' University of St Andrews and RWTH Aachen, 2002, Version 4r3.

[6]

T. Grundhöfer, The groups of projectivities of finite projective and affine planes, in "Eleventh British Combinatorial Conference (London, 1987),'' Ars Combin., 25 (1988), 269-275.

[7]

T. Grundhöfer and P. Müller, Sharply 2-transitive sets of permutations and groups of affine projectivities, Beiträge Algebra Geom., 50 (2009), 143-154.

[8]

M. E. O'Nan, Sharply 2-transitive sets of permutations, in "Proc. Rutgers Group Theory Year, 1983-1984 (New Brunswick, N.J., 1983-1984),'' Cambridge Univ. Press, (1985), 63-67.

[9]

J. Quistorff, A survey on packing and covering problems in the Hamming permutation space, Electron. J. Combin., 13 (2006), 13.

[10]

H. Tarnanen, Upper bounds on permutation codes via linear programming, Europ. J. Combinatorics, 20 (1999), 101-114. doi: 10.1006/eujc.1998.0272.

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