On the Largest intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups

Milad Ahanjideh

doi:10.5802/crmath.320

Combinatoire

On the Largest intersecting set in

{GL}_{2} (q)

and some of its subgroups

Milad Ahanjideh ¹

¹ Department of Industrial Engineering, Boğaziçi University, Istanbul, Turkey

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 497-502.

Résumé

Let $q$ be a power of a prime number and $V$ be a $2$ -dimensional column vector space over a finite field $𝔽_{q}$ . Assume that ${SL}_{2} (V) < G \leq {GL}_{2} (V)$ . In this paper we prove an Erdős–Ko–Rado theorem for intersecting sets of G and we show that every maximum intersecting set of $G$ is either a coset of the stabilizer of a point or a coset of $𝒢_{〈 w 〉}$ , where $𝒢_{〈 w 〉} = {M \in G : \forall v \in V, M v - v \in 〈 w 〉}$ , for some $w \in V ∖ {0}$ . It is also shown that every intersecting set of $G$ is contained in a maximum intersecting set.

Reçu le : 2021-07-16
Révisé le : 2021-12-13
Accepté le : 2021-12-30
Publié le : 2022-05-23

DOI : 10.5802/crmath.320

Classification : 05D05, 20B35

Affiliations des auteurs :

Milad Ahanjideh ¹

¹ Department of Industrial Engineering, Boğaziçi University, Istanbul, Turkey

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Milad Ahanjideh},
     title = {On the {Largest} intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {497--502},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.320},
     language = {en},
}

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Milad Ahanjideh. On the Largest intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 497-502. doi : 10.5802/crmath.320. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.320/

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