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

@article{CRMATH_2022__360_G5_497_0,
     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},
}

TY  - JOUR
AU  - Milad Ahanjideh
TI  - On the Largest intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 497
EP  - 502
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.320
LA  - en
ID  - CRMATH_2022__360_G5_497_0
ER  -

%0 Journal Article
%A Milad Ahanjideh
%T On the Largest intersecting set in ${\protect \rm GL}_2(q)$ and some of its subgroups
%J Comptes Rendus. Mathématique
%D 2022
%P 497-502
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.320
%G en
%F CRMATH_2022__360_G5_497_0

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/

Bibliographie
Cité par

[1] Milad Ahanjideh; Neda Ahanjideh Erdős–Ko–Rado theorem in some linear groups and some projective special linear group, Stud. Sci. Math. Hung., Volume 51 (2014) no. 1, pp. 83-91 | Zbl

[2] Bahman Ahmadi; Karen Meagher The Erdős–Ko–Rado property for some $2$ -transitive groups, Ann. Comb., Volume 19 (2015) no. 4, pp. 621-640 | DOI | Zbl

[3] Bahman Ahmadi; Karen Meagher The Erdős–Ko–Rado property for some permutation groups, Australas. J. Comb., Volume 61 (2015) no. 1, pp. 23-41 | Zbl

[4] Peter J. Cameron; Cheng Yeaw Ku Intersecting families of permutations, Eur. J. Comb., Volume 24 (2003) no. 7, pp. 881-890 | DOI | MR | Zbl

[5] Pál Erdős; Chao Ko; Richard Rado Intersection theorems for systems of finite sets, Q. J. Math, Volume 12 (1961), pp. 313-320 | DOI | MR | Zbl

[6] Peter Frankl; Mikhail Deza On the maximum number of permutations with given maximal or minimal distance, J. Comb. Theory, Volume 22 (1977), pp. 352-360 | DOI | MR | Zbl

[7] Chris Godsil; Karen Meagher A new proof of the Erdős–Ko–Rado theorem for intersecting families of permutations, Eur. J. Comb., Volume 30 (2009) no. 2, pp. 404-414 | DOI | Zbl

[8] Cheng Yeaw Ku; Tony W. H. Wong Intersecting families in the alternating group and direct product of symmetric groups, Electron. J. Comb., Volume 14 (2007) no. 1, R25, 15 pages | MR | Zbl

[9] Benoit Larose; Claudia Malvenuto Stable sets of maximal size in Kneser-type graphs, Eur. J. Comb., Volume 25 (2004) no. 5, pp. 657-673 | DOI | MR | Zbl

[10] Ling Long; Rafael Plaza; Peter Sin; Qing Xiang Characterization of intersecting families of maximum size in $PSL (2, q)$ , J. Comb. Theory, Ser. A, Volume 157 (2018), pp. 461-499 | DOI | MR | Zbl

[11] Karen Meagher An Erdős–Ko–Rado theorem for the group $PSU (3, q)$ , Des. Codes Cryptography, Volume 87 (2019) no. 4, pp. 717-744 | DOI | Zbl

[12] Karen Meagher; Pablo Spiga An Erdős–Ko–Rado theorem for the derangement graph of $PGL (2, q)$ acting on the projective line, J. Comb. Theory, Volume 118 (2011) no. 2, pp. 532-544 | DOI | Zbl

[13] Karen Meagher; Pablo Spiga An Erdős–Ko–Rado-type theorem for the derangement graph of ${PGL}_{3} (q)$ acting on the projective plane,, SIAM J. Discrete Math., Volume 28 (2014) no. 2, pp. 918-941 | DOI | Zbl

[14] Karen Meagher; Pablo Spiga; Pham Huu Tiep An Erdős–Ko–Rado theorem for finite $2$ -transitive groups, Eur. J. Comb., Volume 55 (2016), pp. 100-118 | DOI | Zbl

[15] Pablo Spiga The Erdős–Ko–Rado theorem for the derangement graph of the projective general linear group acting on the projective space, J. Comb. Theory, Volume 166 (2019), pp. 59-90 | DOI | Zbl

Cité par Sources :

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse: