# American Institute of Mathematical Sciences

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.
 [1] Kesong Yan, Qian Liu, Fanping Zeng. Classification of transitive group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5579-5607. doi: 10.3934/dcds.2021089 [2] Robert F. Bailey, John N. Bray. Decoding the Mathieu group M12. Advances in Mathematics of Communications, 2007, 1 (4) : 477-487. doi: 10.3934/amc.2007.1.477 [3] Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699 [4] Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2020, 14 (3) : 423-436. doi: 10.3934/amc.2020058 [5] Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032 [6] Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977 [7] Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115 [8] Bertuel Tangue Ndawa. Infinite lifting of an action of symplectomorphism group on the set of bi-Lagrangian structures. Journal of Geometric Mechanics, 2022  doi: 10.3934/jgm.2022006 [9] Peter Poláčik. On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2657-2667. doi: 10.3934/dcds.2014.34.2657 [10] Wenying Zhang, Zhaohui Xing, Keqin Feng. A construction of bent functions with optimal algebraic degree and large symmetric group. Advances in Mathematics of Communications, 2020, 14 (1) : 23-33. doi: 10.3934/amc.2020003 [11] Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453 [12] Stefano Barbero, Emanuele Bellini, Rusydi H. Makarim. Rotational analysis of ChaCha permutation. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021057 [13] María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021018 [14] Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503 [15] Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9 [16] Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215 [17] Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427 [18] Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041 [19] Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505 [20] Shiping Cao, Anthony Coniglio, Xueyan Niu, Richard H. Rand, Robert S. Strichartz. The mathieu differential equation and generalizations to infinite fractafolds. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1795-1845. doi: 10.3934/cpaa.2020073

2020 Impact Factor: 0.935