Maximal cocliques in the generating graphs of the alternating and symmetric groups
Date
30/03/2022Metadata
Show full item recordAbstract
The generating graph Γ (G) of a finite group G has vertex set the non-identity elements of G, with two elements adjacent exactly when they generate G. A coclique in a graph is an empty induced subgraph, so a coclique in Γ (G) is a subset of G such that no pair of elements generate G. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of G form a coclique in Γ (G), but this coclique need not be maximal. In this paper we determine when the intransitive maximal subgroups of Sn and An are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal in the case of G = An and Sn, when n is prime and n ≠ qd-1/q-1 for all prime powers q and d ≥ 2. Namely, we show that two elements of G have identical sets of neighbours in Γ (G) if and only if they belong to exactly the same maximal subgroups.
Citation
Kelsey , V & Roney-Dougal , C M 2022 , ' Maximal cocliques in the generating graphs of the alternating and symmetric groups ' , Combinatorial Theory , vol. 2 , no. 1 , 56879 . https://doi.org/10.5070/C62156879
Publication
Combinatorial Theory
Status
Peer reviewed
ISSN
2766-1334Type
Journal article
Rights
Copyright © 2022 by the author(s). This work is made available under the terms of a Creative Commons Attribution License, available at https://creativecommons.org/licenses/by/4.0/.
Description
Funding: This work was supported by EPSRC grant number EP/R014604/1. In addition, this work was partially supported by a grant from the Simons Foundation.Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.