Cameron-Liebler sets in permutation groups. (English) Zbl 07921889
Summary: Consider a group \(G\) acting on a set \(\Omega\). A \((G, \Omega)\)-Cameron-Liebler set is a subset of \(G\), whose indicator function is a linear combination of the indicator functions of the cosets of the point stabilizers. We investigate Cameron-Liebler sets in permutation groups, with a focus on constructions of Cameron-Liebler sets for \(2\)-transitive groups.
MSC:
| 20B05 | General theory for finite permutation groups |
References:
| [1] | Ahmadi, Bahman; Meagher, Karen, The Erdős-Ko-Rado property for some permutation groups, Australas. J. Combin., 61, 23-41, 2015 · Zbl 1309.05175 |
| [2] | Ahmadi, Bahman; Meagher, Karen, The Erdős-Ko-Rado property for some 2-transitive groups, Ann. Comb., 19, 4, 621-640, 2015 · Zbl 1326.05068 · doi:10.1007/s00026-015-0285-6 |
| [3] | Bardestani, Mohammad; Mallahi-Karai, Keivan, On the Erdős-Ko-Rado property for finite groups, J. Algebraic Combin., 42, 1, 111-128, 2015 · Zbl 1326.05160 · doi:10.1007/s10801-014-0575-9 |
| [4] | Block, Richard E., On the orbits of collineation groups, Math. Z., 96, 33-49, 1967 · Zbl 0163.42304 · doi:10.1007/BF01111448 |
| [5] | Burnside, W., Theory of groups of finite order, xxiv+512 p. pp., 1955, Dover Publications, Inc., New York · Zbl 0064.25105 |
| [6] | Cameron, P. J.; Liebler, R. A., Tactical decompositions and orbits of projective groups, Linear Algebra Appl., 46, 91-102, 1982 · Zbl 0504.05015 · doi:10.1016/0024-3795(82)90029-5 |
| [7] | Cameron, Peter J.; Ku, C. Y., Intersecting families of permutations, European J. Combin., 24, 7, 881-890, 2003 · Zbl 1026.05001 · doi:10.1016/S0195-6698(03)00078-7 |
| [8] | Cossidente, Antonio; Pavese, Francesco, Cameron-Liebler line classes of \({\rm PG}(3,q)\) admitting \({\rm PGL}(2,q)\), J. Combin. Theory Ser. A, 167, 104-120, 2019 · Zbl 1431.51005 · doi:10.1016/j.jcta.2019.04.004 |
| [9] | De Boeck, M.; D’haeseleer, J., Equivalent definitions for (degree one) Cameron-Liebler classes of generators in finite classical polar spaces, Discrete Math., 343, 1, 13 p. pp., 2020 · Zbl 1443.51002 · doi:10.1016/j.disc.2019.111642 |
| [10] | De Boeck, M.; Rodgers, M.; Storme, L.; Švob, A., Cameron-Liebler sets of generators in finite classical polar spaces, J. Combin. Theory Ser. A, 167, 340-388, 2019 · Zbl 1437.51009 · doi:10.1016/j.jcta.2019.05.005 |
| [11] | De Boeck, M.; Storme, L.; Švob, A., The Cameron-Liebler problem for sets, Discrete Math., 339, 2, 470-474, 2016 · Zbl 1327.05326 · doi:10.1016/j.disc.2015.09.024 |
| [12] | De Bruyn, Bart; Suzuki, Hiroshi, Intriguing sets of vertices of regular graphs, Graphs Combin., 26, 5, 629-646, 2010 · Zbl 1230.05297 · doi:10.1007/s00373-010-0924-y |
| [13] | D’haeseleer, J.; Mannaert, J.; Storme, L.; Švob, A., Cameron-Liebler line classes in AG \((3,q)\), Finite Fields Appl., 67, 17 p. pp., 2020 · Zbl 1459.51002 · doi:10.1016/j.ffa.2020.101706 |
| [14] | Dixon, John D.; Mortimer, Brian, Permutation groups, 163, xii+346 p. pp., 1996, Springer Science & Business Media · Zbl 0951.20001 · doi:10.1007/978-1-4612-0731-3 |
| [15] | Eberhard, Sean, Sharply transitive sets in \({\rm PGL}_2(K)\), Adv. Geom., 21, 4, 611-612, 2021 · Zbl 1476.51005 · doi:10.1515/advgeom-2021-0029 |
| [16] | Ellis, David, Setwise intersecting families of permutations, J. Combin. Theory Ser. A, 119, 4, 825-849, 2012 · Zbl 1237.05210 · doi:10.1016/j.jcta.2011.12.003 |
| [17] | Ellis, David; Friedgut, Ehud; Pilpel, Haran, Intersecting families of permutations, J. Amer. Math. Soc., 24, 3, 649-682, 2011 · Zbl 1285.05171 · doi:10.1090/S0894-0347-2011-00690-5 |
| [18] | Ernst, Alena; Schmidt, Kai-Uwe, Intersection theorems for finite general linear groups, Math. Proc. Cambridge Philos. Soc., 175, 1, 129-160, 2023 · Zbl 1516.05217 · doi:10.1017/S0305004123000075 |
| [19] | Filmus, Y.; Ihringer, F., Boolean degree 1 functions on some classical association schemes, J. Combin. Theory Ser. A, 162, 241-270, 2019 · Zbl 1401.05317 · doi:10.1016/j.jcta.2018.11.006 |
| [20] | Gavrilyuk, A. L.; Metsch, K., A modular equality for Cameron-Liebler line classes, J. Combin. Theory Ser. A, 127, 224-242, 2014 · Zbl 1302.51008 · doi:10.1016/j.jcta.2014.06.004 |
| [21] | Godsil, Christopher; Meagher, Karen, Erdős-Ko-Rado Theorems: Algebraic Approaches, 149, 2016, Cambridge University Press · Zbl 1343.05002 |
| [22] | Hujdurović, Ademir; Kutnar, Klavdija; Marušič, Dragan; Miklavič, Štefko, Intersection density of transitive groups of certain degrees, Algebr. Comb., 5, 2, 289-297, 2022 · Zbl 07524643 · doi:10.5802/alco.209 |
| [23] | Huppert, Bertram, Endliche Gruppen. I, 134, xii+793 p. pp., 1967, Springer-Verlag, Berlin-New York · Zbl 0217.07201 |
| [24] | Isaacs, I. M.; Navarro, Gabriel, Character sums and double cosets, J. Algebra, 320, 10, 3749-3764, 2008 · Zbl 1189.20012 · doi:10.1016/j.jalgebra.2008.08.010 |
| [25] | Isaacs, I. Martin, Character theory of finite groups, No. 69, xii+303 p. pp., 1976, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0337.20005 |
| [26] | King, Oliver H., Surveys in combinatorics 2005, 327, The subgroup structure of finite classical groups in terms of geometric configurations, 29-56, 2005, Cambridge Univ. Press, Cambridge · Zbl 1107.20035 · doi:10.1017/CBO9780511734885.003 |
| [27] | Ku, Cheng Yeaw; Wong, Tony W. H., Intersecting families in the alternating group and direct product of symmetric groups, Electron. J. Combin., 14, 1, 15 p. pp., 2007 · Zbl 1111.05093 · doi:10.37236/943 |
| [28] | Li, Cai-Heng; Pantangi, Venkata Raghu Tej, On the EKR-module property, Algebr. Comb., 7, 2, 577-596, 2024 · Zbl 1536.05445 |
| [29] | Li, Cai Heng; Song, Shu Jiao; Pantangi, Venkata Raghu Tej, Erdős-Ko-Rado problems for permutation groups, 2020 |
| [30] | Long, Ling; Plaza, Rafael; Sin, Peter; Xiang, Qing, Characterization of intersecting families of maximum size in \(PSL(2, q)\), J. Combin. Theory Ser. A, 157, 461-499, 2018 · Zbl 1385.05070 · doi:10.1016/j.jcta.2018.03.006 |
| [31] | Meagher, Karen, An Erdős-Ko-Rado theorem for the group \({P}{S}{U}(3,q)\), Des. Codes Cryptogr., 87, 4, 717-744, 2019 · Zbl 1407.05232 · doi:10.1007/s10623-018-0537-7 |
| [32] | Meagher, Karen; Razafimahatratra, Andriaherimanana Sarobidy, Some Erdős-Ko-Rado results for linear and affine groups of degree two, Art Discrete Appl. Math., 6, 1, 30 p. pp., 2023 · Zbl 1502.05091 · doi:10.26493/2590-9770.1494.1e4 |
| [33] | Meagher, Karen; Sin, Peter, All 2-transitive groups have the EKR-module property, J. Combin. Theory Ser. A, 177, 21 p. pp., 2021 · Zbl 1448.05228 · doi:10.1016/j.jcta.2020.105322 |
| [34] | Meagher, Karen; Spiga, Pablo, An Erdős-Ko-Rado theorem for the derangement graph of PGL(2, q) acting on the projective line, J. Combin. Theory Ser. A, 118, 2, 532-544, 2011 · Zbl 1227.05163 · doi:10.1016/j.jcta.2010.11.003 |
| [35] | Meagher, Karen; Spiga, Pablo; Tiep, Pham Huu, An Erdős-Ko-Rado theorem for finite 2-transitive groups, European J. Combin., 55, 100-118, 2016 · Zbl 1333.05306 · doi:10.1016/j.ejc.2016.02.005 |
| [36] | Metsch, K., An improved bound on the existence of Cameron-Liebler line classes, J. Combin. Theory Ser. A, 121, 89-93, 2014 · Zbl 1284.51008 · doi:10.1016/j.jcta.2013.09.004 |
| [37] | Mogilnykh, Ivan, Completely regular codes in Johnson and Grassmann graphs with small covering radii, Electron. J. Combin., 29, 2, 15 p. pp., 2022 · Zbl 1492.05022 · doi:10.37236/10083 |
| [38] | Palmarin, Daniel Michael, Cameron-Liebler Sets for 2-Transitive Groups, 2020 |
| [39] | Piatetski-Shapiro, Ilya, Complex representations of \({\rm GL}(2,\,K)\) for finite fields \(K\), 16, vii+71 p. pp., 1983, American Mathematical Society, Providence, RI · Zbl 0513.20026 |
| [40] | Sagan, Bruce E., The symmetric group: Representations, combinatorial algorithms, and symmetric functions, xviii+197 p. pp., 1991, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA · Zbl 0823.05061 |
| [41] | Spiga, Pablo, The Erdős-Ko-Rado theorem for the derangement graph of the projective general linear group acting on the projective space, J. Combin. Theory Ser. A, 166, 59-90, 2019 · Zbl 1416.05276 · doi:10.1016/j.jcta.2019.02.015 |
| [42] | Wang, Jun; Zhang, Sophia J., An Erdős-Ko-Rado-type theorem in Coxeter groups, European J. Combin., 29, 5, 1112-1115, 2008 · Zbl 1140.20001 · doi:10.1016/j.ejc.2007.07.002 |
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