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{
    "pk": 64824,
    "title": "Maximal cocliques in the generating graphs of the alternating and symmetric groups",
    "subtitle": null,
    "abstract": "The generating graph $\\Gamma(G)$ 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 $\\Gamma(G)$ 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 $\\Gamma(G)$, but this coclique need not be maximal.  In this paper we determine when the intransitive maximal subgroups of $\\mathrm{S}_n$ and $\\mathrm{A}_n$ are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal in the case of $G = \\mathrm{A}_n$ and $\\mathrm{S}_n$, when $n$ is prime and ${n \\neq \\frac{q^d-1}{q-1}}$ for all prime powers $q$ and $d \\geq 2$. Namely, we show that two elements of $G$ have identical sets of neighbours in $\\Gamma(G)$ if and only if they belong to exactly the same maximal subgroups.\n \nMathematics Subject Classifications: 20D06, 05C25, 20B35",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Generating graph"
        },
        {
            "word": "alternating groups"
        },
        {
            "word": "symmetric groups"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/6152c771",
    "frozenauthors": [
        {
            "first_name": "Veronica",
            "middle_name": "",
            "last_name": "Kelsey",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of Manchester, U.K.",
            "department": ""
        },
        {
            "first_name": "Colva",
            "middle_name": "M.",
            "last_name": "Roney-Dougal",
            "name_suffix": "",
            "institution": "School of Mathematics and Statistics, University of St Andrews, U.K.",
            "department": ""
        }
    ],
    "date_submitted": "2022-03-23T20:08:02Z",
    "date_accepted": "2022-03-23T20:08:02Z",
    "date_published": "2022-03-31T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64824/galley/49634/download/"
        }
    ]
}