# ALGEBRAIC COMBINATORICS

Intersection density of transitive groups of certain degrees
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 $v\in V$. More generally, a subset $ℱ$ of $G$ is an intersecting set if every pair of elements of $ℱ$ is intersecting. The intersection density $\rho \left(G\right)$ 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 $v\in V$. 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 $1$.

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
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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/

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