ALGEBRAIC COMBINATORICS

no. 2
Intersection density of transitive groups of certain degrees
Hujdurović, Ademir1; Kutnar, Klavdija1; Marušič, Dragan2, 3; Miklavič, Štefko2, 4
1 University of Primorska UP IAM & UP FAMNIT Glagoljaška 8, 6000 Koper Slovenia
2 IMFM Jadranska 19, 1000 Ljubljana Slovenia
3 University of Primorska, UP IAM & UP FAMNIT Glagoljaška 8, 6000 Koper Slovenia
4 University of Primorska, UP IAM & UP FAMNIT, Glagoljaška 8, 6000 Koper Slovenia
Algebraic Combinatorics, Volume 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 v g =v h for some v∈V. More generally, a subset ℱ of G is an intersecting set if every pair of elements of ℱ is intersecting. The intersection density ρ(G) of a transitive permutation group G is the maximum value of the quotient |ℱ|/|G v | where ℱ runs over all intersecting sets in G and G v is a stabilizer of v∈V. 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.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.209
Classification: 05C25, 20B25
Keywords: intersection density, derangement, derangement graph, transitive permutation group
Hujdurović, Ademir 1; Kutnar, Klavdija 1; Marušič, Dragan 2, 3; Miklavič, Štefko 2, 4

1 University of Primorska UP IAM & UP FAMNIT Glagoljaška 8, 6000 Koper Slovenia
2 IMFM Jadranska 19, 1000 Ljubljana Slovenia
3 University of Primorska, UP IAM & UP FAMNIT Glagoljaška 8, 6000 Koper Slovenia
4 University of Primorska, UP IAM & UP FAMNIT, Glagoljaška 8, 6000 Koper Slovenia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2022__5_2_289_0,
     author = {Hujdurovi\'c, Ademir and Kutnar, Klavdija and Maru\v{s}i\v{c}, Dragan and Miklavi\v{c}, \v{S}tefko},
     title = {Intersection density of transitive groups of certain degrees},
     journal = {Algebraic Combinatorics},
     pages = {289--297},
     publisher = {The Combinatorics Consortium},
     volume = {5},
     number = {2},
     year = {2022},
     doi = {10.5802/alco.209},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.209/}
}
TY  - JOUR
AU  - Hujdurović, Ademir
AU  - Kutnar, Klavdija
AU  - Marušič, Dragan
AU  - Miklavič, Štefko
TI  - Intersection density of transitive groups of certain degrees
JO  - Algebraic Combinatorics
PY  - 2022
SP  - 289
EP  - 297
VL  - 5
IS  - 2
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.209/
DO  - 10.5802/alco.209
LA  - en
ID  - ALCO_2022__5_2_289_0
ER  - 
%0 Journal Article
%A Hujdurović, Ademir
%A Kutnar, Klavdija
%A Marušič, Dragan
%A Miklavič, Štefko
%T Intersection density of transitive groups of certain degrees
%J Algebraic Combinatorics
%D 2022
%P 289-297
%V 5
%N 2
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.209/
%R 10.5802/alco.209
%G en
%F ALCO_2022__5_2_289_0
Hujdurović, Ademir; Kutnar, Klavdija; Marušič, Dragan; Miklavič, Štefko. Intersection density of transitive groups of certain degrees. Algebraic Combinatorics, Volume 5 (2022) no. 2, pp. 289-297. doi : 10.5802/alco.209. https://alco.centre-mersenne.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), Paper no. 105390, 26 pages | DOI | MR | Zbl

[13] Razafimahatratra, Andriaherimanana Sarobidy On complete multipartite derangement graphs, Ars Math. Contemp., Volume 21 (2021) no. 1, Paper no. 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

Cited by Sources: