Intersection density of transitive groups of certain degrees
Algebraic Combinatorics, Tome 5 (2022) no. 2, pp. 289-297.

Two elements g and h of a permutation group G acting on a set V are said to be intersecting if vg=vh for some vV. More generally, a subset of G is an intersecting set if every pair of elements of is intersecting. The intersection density ρ(G) of a transitive permutation group G is the maximum value of the quotient ||/|Gv| where runs over all intersecting sets in G and Gv is a stabilizer of vV. In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be either 1 or 2. In addition, it is proved that the intersection density of transitive groups of prime power degree is 1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/alco.209
Classification : 05C25, 20B25
Mots-clés : intersection density, derangement, derangement graph, transitive permutation group
Hujdurović, Ademir 1 ; Kutnar, Klavdija 1 ; Marušič, Dragan 2, 3 ; Miklavič, Štefko 4, 3

1 University of Primorska UP IAM & UP FAMNIT Glagoljaška 8, 6000 Koper Slovenia
2 University of Primorska, UP IAM & UP FAMNIT Glagoljaška 8, 6000 Koper Slovenia
3 IMFM Jadranska 19, 1000 Ljubljana Slovenia
4 University of Primorska, UP IAM & UP FAMNIT, Glagoljaška 8, 6000 Koper Slovenia
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Hujdurović, Ademir; Kutnar, Klavdija; Marušič, Dragan; Miklavič, Štefko. Intersection density of transitive groups of certain degrees. Algebraic Combinatorics, Tome 5 (2022) no. 2, pp. 289-297. doi : 10.5802/alco.209. http://www.numdam.org/articles/10.5802/alco.209/

[1] Fein, Burton; Kantor, William M.; Schacher, Murray Relative Brauer groups. II, J. Reine Angew. Math., Volume 328 (1981), pp. 39-57 | DOI | MR | Zbl

[2] Godsil, Chris; Meagher, Karen Erdos–Ko–Rado theorems: algebraic approaches, Cambridge Studies in Advanced Mathematics, 149, Cambridge University Press, Cambridge, 2016, xvi+335 pages | DOI | MR | Zbl

[3] Itô, Noboru Transitive permutation groups of degree p=2q+1,p and q being prime numbers, Bull. Amer. Math. Soc., Volume 69 (1963), pp. 165-192 | DOI | MR | Zbl

[4] Itô, Noboru Transitive permutation groups of degree p=2q+1, p and q being prime numbers. II, Trans. Amer. Math. Soc., Volume 113 (1964), pp. 454-487 | DOI | MR | Zbl

[5] Itô, Noboru Transitive permutation groups of degree p=2q+1,p and q being prime numbers. III, Trans. Amer. Math. Soc., Volume 116 (1965), pp. 151-166 | DOI | MR | Zbl

[6] Itô, Noboru; Wada, Tomoyuki A note on transitive permutation groups of degree 2p, Tensor (N.S.), Volume 26 (1972), pp. 105-106 | MR | Zbl

[7] Jordan, Camille Recherches sur les substitutions, J. Math. Pures Appl., Volume 17 (1872), pp. 351-367 | Zbl

[8] Li, Cai Heng; Song, Shu Jiao; Pantangi, Venkata Raghu Tej Erdos–Ko–Rado problems for permutation groups (2021) (https://arxiv.org/abs/2006.10339)

[9] Liebeck, Martin W.; Saxl, Jan Primitive permutation groups containing an element of large prime order, J. London Math. Soc. (2), Volume 31 (1985) no. 2, pp. 237-249 | DOI | MR | Zbl

[10] Marušič, Dragan On vertex symmetric digraphs, Discrete Math., Volume 36 (1981) no. 1, pp. 69-81 | DOI | MR | Zbl

[11] Marušič, Dragan; Potočnik, Primož Semisymmetry of generalized Folkman graphs, European J. Combin., Volume 22 (2001) no. 3, pp. 333-349 | DOI | MR | Zbl

[12] Meagher, Karen; Razafimahatratra, Andriaherimanana Sarobidy; Spiga, Pablo On triangles in derangement graphs, J. Combin. Theory Ser. A, Volume 180 (2021), 105390, 26 pages | DOI | MR | Zbl

[13] Razafimahatratra, Andriaherimanana Sarobidy On complete multipartite derangement graphs, Ars Math. Contemp., Volume 21 (2021) no. 1, P1.07, 15 pages | DOI | MR | Zbl

[14] Scott, Leonard L. On permutation groups of degree 2p, Math. Z., Volume 126 (1972), pp. 227-229 | DOI | MR | Zbl

[15] Wielandt, Helmut Primitive Permutationsgruppen vom Grad 2p, Math. Z., Volume 63 (1956), pp. 478-485 | DOI | MR | Zbl

[16] Wielandt, Helmut Finite permutation groups, Academic Press, New York-London, 1964, x+114 pages (Translated from the German by R. Bercov) | MR | Zbl

  • Maleki, Roghayeh; Razafimahatratra, Andriaherimanana Sarobidy The intersection density of non-quasiprimitive groups of degree 3p, Discrete Mathematics, Volume 348 (2025) no. 4, p. 114364 | DOI:10.1016/j.disc.2024.114364
  • Hujdurović, Ademir; Kovács, István; Kutnar, Klavdija; Marušič, Dragan Intersection density of transitive groups with small cyclic point stabilizers, European Journal of Combinatorics, Volume 124 (2025), p. 13 (Id/No 104079) | DOI:10.1016/j.ejc.2024.104079 | Zbl:7960749
  • D'haeseleer, Jozefien; Meagher, Karen; Pantangi, Venkata Raghu Tej Cameron-Liebler sets in permutation groups, Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 1157-1182 | DOI:10.5802/alco.363 | Zbl:7921889
  • Meagher, Karen; Razafimahatratra, Andriaherimanana Sarobidy On the intersection density of the Kneser graph K(n,3), European Journal of Combinatorics, Volume 118 (2024), p. 15 (Id/No 103910) | DOI:10.1016/j.ejc.2023.103910 | Zbl:1535.05134
  • Behajaina, Angelot; Maleki, Roghayeh; Razafimahatratra, Andriaherimanana Sarobidy On the intersection spectrum of PSL2(q), Journal of Algebraic Combinatorics, Volume 60 (2024) no. 4, p. 899 | DOI:10.1007/s10801-024-01356-5
  • Behajaina, Angelot; Maleki, Roghayeh; Razafimahatratra, Andriaherimanana Sarobidy Intersection density of imprimitive groups of degree pq, Journal of Combinatorial Theory. Series A, Volume 208 (2024), p. 23 (Id/No 105922) | DOI:10.1016/j.jcta.2024.105922 | Zbl:7920037
  • Kutnar, Klavdija; Marušič, Dragan; Pujol, Cyril Intersection density of cubic symmetric graphs, Journal of Algebraic Combinatorics, Volume 57 (2023) no. 4, pp. 1313-1326 | DOI:10.1007/s10801-023-01228-4 | Zbl:1516.05090
  • Razafimahatratra, Andriaherimanana Sarobidy On the intersection density of primitive groups of degree a product of two odd primes, Journal of Combinatorial Theory. Series A, Volume 194 (2023), p. 31 (Id/No 105707) | DOI:10.1016/j.jcta.2022.105707 | Zbl:1521.20003
  • Behajaina, Angelot; Maleki, Roghayeh; Razafimahatratra, Andriaherimanana Sarobidy On the intersection density of the symmetric group acting on uniform subsets of small size, Linear Algebra and its Applications, Volume 664 (2023), pp. 61-103 | DOI:10.1016/j.laa.2023.01.010 | Zbl:1509.05170
  • Hujdurović, Ademir; Kovács, István; Kutnar, Klavdija; Marušič, Dragan Intersection density of transitive groups with cyclic point stabilizers, arXiv (2022) | DOI:10.48550/arxiv.2201.11015 | arXiv:2201.11015
  • Behajaina, Angelot; Maleki, Roghayeh; Sarobidy Razafimahatratra, Andriaherimanana On the intersection density of the symmetric group acting on uniform subsets of small size, arXiv (2022) | DOI:10.48550/arxiv.2201.09727 | arXiv:2201.09727
  • Hujdurović, Ademir; Kutnar, Klavdija; Marušič, Dragan; Miklavič, Štefko On maximum intersecting sets in direct and wreath product of groups, arXiv (2021) | DOI:10.48550/arxiv.2108.03943 | arXiv:2108.03943

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