Abelian sections of the symmetric groups with respect to their index View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2021-10-16

AUTHORS

Luca Sabatini

ABSTRACT

We show the existence of an absolute constant α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} such that, for every k≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 3$$\end{document}, G:=Sym(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G:= \mathop {\mathrm {Sym}}(k)$$\end{document}, and for every H⩽G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \leqslant G$$\end{document} of index at least 3, one has |H/H′|≤|G:H|α/loglog|G:H|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|H/H'| \le |G:H|^{\alpha / \log \log |G:H|}$$\end{document}. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups. More... »

PAGES

3-12

References to SciGraph publications

  • 1991-05. Nilpotent subgroups of finite soluble groups in ARCHIV DER MATHEMATIK
  • 1997. How Abelian is a Finite Group? in THE MATHEMATICS OF PAUL ERDÖS I
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00013-021-01667-0

    DOI

    http://dx.doi.org/10.1007/s00013-021-01667-0

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1141940618


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Universit\u00e0 degli studi di Firenze, Firenze, Italy", 
              "id": "http://www.grid.ac/institutes/grid.8404.8", 
              "name": [
                "Universit\u00e0 degli studi di Firenze, Firenze, Italy"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Sabatini", 
            "givenName": "Luca", 
            "id": "sg:person.014226040231.16", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014226040231.16"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf01200083", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1017348309", 
              "https://doi.org/10.1007/bf01200083"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-642-60408-9_27", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041451025", 
              "https://doi.org/10.1007/978-3-642-60408-9_27"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2021-10-16", 
        "datePublishedReg": "2021-10-16", 
        "description": "We show the existence of an absolute constant \u03b1>0\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha >0$$\\end{document} such that, for every k\u22653\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k \\ge 3$$\\end{document}, G:=Sym(k)\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$G:= \\mathop {\\mathrm {Sym}}(k)$$\\end{document}, and for every H\u2a7dG\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H \\leqslant G$$\\end{document} of index at least 3, one has |H/H\u2032|\u2264|G:H|\u03b1/loglog|G:H|\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|H/H'| \\le |G:H|^{\\alpha / \\log \\log |G:H|}$$\\end{document}. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s00013-021-01667-0", 
        "inLanguage": "en", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1052783", 
            "issn": [
              "0003-889X", 
              "1420-8938"
            ], 
            "name": "Archiv der Mathematik", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "1", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "118"
          }
        ], 
        "keywords": [
          "group", 
          "index", 
          "family", 
          "sections", 
          "respect", 
          "existence", 
          "inequality", 
          "symmetric group", 
          "large finite groups", 
          "finite group"
        ], 
        "name": "Abelian sections of the symmetric groups with respect to their index", 
        "pagination": "3-12", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1141940618"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00013-021-01667-0"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00013-021-01667-0", 
          "https://app.dimensions.ai/details/publication/pub.1141940618"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-05-20T07:39", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220519/entities/gbq_results/article/article_908.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s00013-021-01667-0"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00013-021-01667-0'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00013-021-01667-0'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00013-021-01667-0'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00013-021-01667-0'


     

    This table displays all metadata directly associated to this object as RDF triples.

    76 TRIPLES      22 PREDICATES      37 URIs      27 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00013-021-01667-0 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N1908324e89ed4b6bb9d34baa0cc76a23
    4 schema:citation sg:pub.10.1007/978-3-642-60408-9_27
    5 sg:pub.10.1007/bf01200083
    6 schema:datePublished 2021-10-16
    7 schema:datePublishedReg 2021-10-16
    8 schema:description We show the existence of an absolute constant α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} such that, for every k≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 3$$\end{document}, G:=Sym(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G:= \mathop {\mathrm {Sym}}(k)$$\end{document}, and for every H⩽G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H \leqslant G$$\end{document} of index at least 3, one has |H/H′|≤|G:H|α/loglog|G:H|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|H/H'| \le |G:H|^{\alpha / \log \log |G:H|}$$\end{document}. This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.
    9 schema:genre article
    10 schema:inLanguage en
    11 schema:isAccessibleForFree true
    12 schema:isPartOf N6e83e01625dc42c185d293429f9c09a2
    13 Nbeb0df88369d49129074760ce6d0d1d2
    14 sg:journal.1052783
    15 schema:keywords existence
    16 family
    17 finite group
    18 group
    19 index
    20 inequality
    21 large finite groups
    22 respect
    23 sections
    24 symmetric group
    25 schema:name Abelian sections of the symmetric groups with respect to their index
    26 schema:pagination 3-12
    27 schema:productId N3d9563e1678745f9af4e4f1c286847ea
    28 Nfe9c8311edec4c96900fb48aaf9ff686
    29 schema:sameAs https://app.dimensions.ai/details/publication/pub.1141940618
    30 https://doi.org/10.1007/s00013-021-01667-0
    31 schema:sdDatePublished 2022-05-20T07:39
    32 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    33 schema:sdPublisher N5f03d35511034f19aa1f88cb10845ec7
    34 schema:url https://doi.org/10.1007/s00013-021-01667-0
    35 sgo:license sg:explorer/license/
    36 sgo:sdDataset articles
    37 rdf:type schema:ScholarlyArticle
    38 N1908324e89ed4b6bb9d34baa0cc76a23 rdf:first sg:person.014226040231.16
    39 rdf:rest rdf:nil
    40 N3d9563e1678745f9af4e4f1c286847ea schema:name doi
    41 schema:value 10.1007/s00013-021-01667-0
    42 rdf:type schema:PropertyValue
    43 N5f03d35511034f19aa1f88cb10845ec7 schema:name Springer Nature - SN SciGraph project
    44 rdf:type schema:Organization
    45 N6e83e01625dc42c185d293429f9c09a2 schema:issueNumber 1
    46 rdf:type schema:PublicationIssue
    47 Nbeb0df88369d49129074760ce6d0d1d2 schema:volumeNumber 118
    48 rdf:type schema:PublicationVolume
    49 Nfe9c8311edec4c96900fb48aaf9ff686 schema:name dimensions_id
    50 schema:value pub.1141940618
    51 rdf:type schema:PropertyValue
    52 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    53 schema:name Mathematical Sciences
    54 rdf:type schema:DefinedTerm
    55 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    56 schema:name Pure Mathematics
    57 rdf:type schema:DefinedTerm
    58 sg:journal.1052783 schema:issn 0003-889X
    59 1420-8938
    60 schema:name Archiv der Mathematik
    61 schema:publisher Springer Nature
    62 rdf:type schema:Periodical
    63 sg:person.014226040231.16 schema:affiliation grid-institutes:grid.8404.8
    64 schema:familyName Sabatini
    65 schema:givenName Luca
    66 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014226040231.16
    67 rdf:type schema:Person
    68 sg:pub.10.1007/978-3-642-60408-9_27 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041451025
    69 https://doi.org/10.1007/978-3-642-60408-9_27
    70 rdf:type schema:CreativeWork
    71 sg:pub.10.1007/bf01200083 schema:sameAs https://app.dimensions.ai/details/publication/pub.1017348309
    72 https://doi.org/10.1007/bf01200083
    73 rdf:type schema:CreativeWork
    74 grid-institutes:grid.8404.8 schema:alternateName Università degli studi di Firenze, Firenze, Italy
    75 schema:name Università degli studi di Firenze, Firenze, Italy
    76 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...