ISSN 2590-9770
The Art of Discrete and Applied Mathematics 6 (2023) #P1.05
https://doi.org/10.26493/2590-9770.1494.1e4
(Also available at http://adam-journal.eu)
Some Erdős-Ko-Rado results for linear and
affine groups of degree two*
Karen Meagher† , Andriaherimanana Sarobidy Razafimahatratra
Department of Mathematics and Statistics, University of Regina, Regina,
Saskatchewan S4S 0A2, Canada
Received 16 October 2021, accepted 2 February 2022, published online 4 October 2022
Abstract
In this paper, we show that both the general linear group GL(2, q) and the special linear
group SL(2, q) have both the EKR property and the EKR-module property. This is done
using an algebraic method; a weighted adjacency matrix for the derangement graph for the
group is found and Hoffman’s ratio bound is applied to this matrix. We also consider the
group AGL(2, q) and the 2-intersecting sets in PGL(2, q).
Keywords: derangement graph, independent sets, Erdős-Ko-Rado Theorem, Symmetric group, Gen-
eral linear group, Affine linear group, Projective linear group
Math. Subj. Class.: Primary 05C35; Secondary 05C69, 20B05
*We are grateful to the anonymous referee for their feedback.
†Corresponding author. Research supported in part by an NSERC Discovery Research Grant, Application No.:
RGPIN-2018-03952.
E-mail addresses: meagherk@uregina.ca (Karen Meagher), sarobidy@phystech.edu (Andriaherimanana
Sarobidy Razafimahatratra)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 6 (2023) #P1.05
https://doi.org/10.26493/2590-9770.1494.1e4
(Dostopno tudi na http://adam-journal.eu)
Nekaj rezultatov Erdős-Ko-Radovega tipa za
linearne in afine grupe stopnje dve*
Karen Meagher† , Andriaherimanana Sarobidy Razafimahatratra
Department of Mathematics and Statistics, University of Regina, Regina,
Saskatchewan S4S 0A2, Canada
Prejeto 16. oktobra 2021, sprejeto 2. februarja 2022, objavljeno na spletu 4. oktobra 2022
Povzetek
V tem članku pokažemo, da imata tako splošna linearna grupa GL(2, q) kot tudi posebna
linearna grupa SL(2, q) lastnost EKR in EKR-modulsko lastnost. To storimo z uporabo al-
gebraične metode; poiščemo uteženo matriko sosednosti grafa premestitev (t.j. permutacij
brez negibnih točk) grupe in za to matriko uporabimo Hoffmanovo razmerje. Obravnavamo
tudi grupo AGL(2, q) ter 2-presečne množice v PGL(2, q).
Ključne besede: Graf premestitev, neodvisne množice, Erdős-Ko-Radov izrek, simetrična grupa, splošna
linearna grupa, afina linearna grupa, projektivna linearna grupa.
Math. Subj. Class.: Primarna klasifikacija: 05C35; sekundarna klasifikacija 05C69, 20B05
*Hvaležni smo recenzentom za njihovo povratno informacijo.
†Kontaktni avtor. Raziskava je delno podprta s strani NSERC Raziskovalne štipendije Discovery, št.: RGPIN-
2018-03952.
E-poštni naslovi: meagherk@uregina.ca (Karen Meagher), sarobidy@phystech.edu (Andriaherimanana
Sarobidy Razafimahatratra)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/