Finite locally-quasiprimitive graphs
type: Item
instance of: scholarly article
title: Finite locally-quasiprimitive graphs
zbMATH Open document ID: 0996.05068
DOI: 10.1016/S0012-365X(01)00258-8
author: Cai Heng Li
author: Cheryl E. Praeger
author: Akshay Venkatesh
author: Sanming Zhou
published in: Discrete Mathematics
publication date: 2002-05-15T00:00:00Z
review text: A quasiprimitive permutation group contains no non-trivial non-transitive normal subgroups. A finite graph is said to be locally quasiprimitive with respect to a subgroup of automorphisms \(G\) if each vertex-stabilizer \(G_{\alpha}\) acts quasiprimitively on the neighbbors of \( \alpha \). An infinite family of locally-quasiprimitive but not locally-primitive graph (with respect to the full automorphism group) based on \( \text{PSL}(2,p)\)s is constructed in the paper; hence, the concept of local quasiprimitivity is shown to be weaker than the concept of local primitivity. The family of locally-quasiprimitive graphs is shown to be closed under the formation of the so-called normal quotients, induced by a normal subgroup, and each locally-quasiprimitive graph is shown to be a multicover of each of its normal quotients. This allows for the concept of minimal locally-quasiprimitive graphs -- the ``building stones'' of all graphs in this class -- which gives hope for possible classification of all locally-quasiprimitive graphs. Both the structure of these minimal graphs as well as the process of obtaining all locally-quasiprimitive graphs whose normal quotient is a given locally-quasiprimitive graph are considered. To facilitate the second task, the concept of an extender of a locally-quasiprimitive graph is introduced, and the simple \(G\)-admissible covers of such an extender are shown to correspond to locally-quasiprimitive normal multicovers of the original graph. The paper closes with some partial results on the problem of reconstructing a locally-quasiprimitive graph from its normal quotients, and with an extensive list of open problems.
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1741708
zbMATH Keywords: arc-transitive graph
zbMATH Keywords: quasiprimitive action
zbMATH Keywords: automorphism group
zbMATH Keywords: stabilizer
zbMATH Keywords: normal quotient
reviewed by: Robert Jajcay
Wikidata QID: Q56987944
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1741708