A class of graphs arising from the action of \(\text{PSL}(2,q^ 2)\) on cosets of \(\text{PGL}(2,q)\)
type: Item
instance of: scholarly article
title: A class of graphs arising from the action of \(\text{PSL}(2,q^ 2)\) on cosets of \(\text{PGL}(2,q)\)
zbMATH Open document ID: 0811.05030
DOI: 10.1016/0012-365X(93)E0065-C
author: Dragan Marušič
author: Raffaele Scapellato
published in: Discrete Mathematics
publication date: 1995-01-16T00:00:00Z
review text: Motivated by the study of vertex-transitive graphs (those with a vertex- transitive automorphism group), this paper first gives a detailed description of the cosets of \(\text{PSL} (2,q)\) in \(\text{PSL}(2,q^ 2)\) (for any odd prime power \(q\)). That description is then used to describe the action of \(\text{PSL}(2,q)\) on these cosets and on the cosets of a subgroup isomorphic to \(\text{PGL}(2,q)\) which is maximal in \(\text{PSL}(2, q^ 2)\). Specifically the paper gives a detailed description of all suborbits (orbits of a point-stabilizer subgroup) of this primitive permutation action. The proofs use an interesting collection of elementary linear algebra calculations with \(2\times 2\) matrices over \(\text{GF}(q^ 2)\). (There seems to be a misprint in the top right entry of Table 1, which should say \(q(q + 1)\) instead of \(q(q - 1)\), and a few misprints in Lemma 4.1 (i).) The paper also briefly describes how each of these suborbits gives rise to an undirected vertex- transitive graph which is not a Cayley graph.
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20B99
zbMATH DE Number: 701686
zbMATH Keywords: vertex-transitive graphs
zbMATH Keywords: cosets
zbMATH Keywords: suborbits
reviewed by: Albert J. Goodman
MaRDI profile type: Publication
cites work: Q3050438
cites work: Distance-transitive representations of groups G with \(PSL_ 2(q)\trianglelefteq G\leq P\Gamma L_ 2(q)\)
cites work: HOW TO DESCRIBE A GRAPH
cites work: Primitive Permutation Groups Containing an Element of Large Prime Order
cites work: Q5653952
cites work: Classifying vertex-transitive graphs whose order is a product of two primes
cites work: Vertex-primitive graphs of order a product of two distinct primes
full work available at URL: 0012-365x(93)e0065-c
OpenAlex ID: W2087261991
Recommended article: Expansion of coset graphs of \(\mathrm{PSL}_2(\mathrm{F}_p)\)
Recommended article: Ramanujan graphs on cosets of \(\operatorname{PGL}_2(\mathbb F_q)\)
Recommended article: A class of non-Cayley vertex-transitive graphs associated with PSL(2,\(p\))
Recommended article: Q4405802
Recommended article: An Erdős-Ko-Rado theorem for the derangement graph of PGL(\(2,q\)) acting on the projective line
Recommended article: Classifying and generating exact coset representatives of \(\operatorname{PGL}_2(\mathbb{F}_q)\) in \(\operatorname{PGL}_2(\mathbb{F}_{q^2})\)
Recommended article: On a Graph Related to Subgroups Generated by a Conjugacy Class in Finite Groups
Recommended article: ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES
Recommended article: On a Graph Related to Conjugacy Classes of Groups
Recommended article: Some applications of pq-groups in graph theory
description: scientific article; zbMATH DE number 701686