Products of powers in finite simple groups View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1996-06

AUTHORS

C. Martinez, E. Zelmanov

ABSTRACT

LetG be a group. For a natural numberd≥1 letGd denote the subgroup ofG generated by all powersad,a∈G. A. Shalev raised the question if there exists a functionN=N(m, d) such that for anm-generated finite groupG an arbitrary element fromGd can be represented asa1d...aNd,ai∈G. The positive answer to this question would imply that in a finitely generated profinite groupG all power subgroupsGd are closed and that an arbitrary subgroup of finite index inG is closed. In [5,6] the first author proved the existence of such a function for nilpotent groups and for finite solvable groups of bounded Fitting height. Another interpretation of the existence ofN(m, d) is definability of power subgroupsGd (see [10]). In this paper we address the question for finite simple groups. All finite simple groups are known to be 2-generated. Thus, we prove the following: THEOREM:There exists a function N=N(d) such that for an arbitrary finite simple group G either Gd=1 orG={a1d...aNd|ai∈G}. The proof is based on the Classification of finite simple groups and sometimes resorts to a case-by-case analysis. More... »

PAGES

469-479

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf02937318

DOI

http://dx.doi.org/10.1007/bf02937318

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1038620902


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/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }, 
      {
        "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"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "University of Oviedo", 
          "id": "https://www.grid.ac/institutes/grid.10863.3c", 
          "name": [
            "Departamento de Matem\u00e1ticas, Universidad de Oviedo, 33.007, Oviedo, Spain"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Martinez", 
        "givenName": "C.", 
        "id": "sg:person.015261576461.61", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015261576461.61"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "University of Chicago", 
          "id": "https://www.grid.ac/institutes/grid.170205.1", 
          "name": [
            "Department of Mathematics, University of Chicago, 60637, Chicago, IL, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Zelmanov", 
        "givenName": "E.", 
        "id": "sg:person.01264523514.92", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01264523514.92"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "https://doi.org/10.1090/s0002-9947-1994-1264149-8", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1006073338"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.1016/0097-3165(72)90102-1", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1017584976"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "https://doi.org/10.2307/2155003", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1069793668"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "1996-06", 
    "datePublishedReg": "1996-06-01", 
    "description": "LetG be a group. For a natural numberd\u22651 letGd denote the subgroup ofG generated by all powersad,a\u2208G. A. Shalev raised the question if there exists a functionN=N(m, d) such that for anm-generated finite groupG an arbitrary element fromGd can be represented asa1d...aNd,ai\u2208G. The positive answer to this question would imply that in a finitely generated profinite groupG all power subgroupsGd are closed and that an arbitrary subgroup of finite index inG is closed. In [5,6] the first author proved the existence of such a function for nilpotent groups and for finite solvable groups of bounded Fitting height. Another interpretation of the existence ofN(m, d) is definability of power subgroupsGd (see [10]). In this paper we address the question for finite simple groups. All finite simple groups are known to be 2-generated. Thus, we prove the following: THEOREM:There exists a function N=N(d) such that for an arbitrary finite simple group G either Gd=1 orG={a1d...aNd|ai\u2208G}. The proof is based on the Classification of finite simple groups and sometimes resorts to a case-by-case analysis.", 
    "genre": "research_article", 
    "id": "sg:pub.10.1007/bf02937318", 
    "inLanguage": [
      "en"
    ], 
    "isAccessibleForFree": false, 
    "isPartOf": [
      {
        "id": "sg:journal.1136632", 
        "issn": [
          "0021-2172", 
          "1565-8511"
        ], 
        "name": "Israel Journal of Mathematics", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "2", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "96"
      }
    ], 
    "name": "Products of powers in finite simple groups", 
    "pagination": "469-479", 
    "productId": [
      {
        "name": "readcube_id", 
        "type": "PropertyValue", 
        "value": [
          "b7f8bb3645ff8f823b34e78913131e40c19227c5dd999035fcdc3234936d3afb"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/bf02937318"
        ]
      }, 
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1038620902"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/bf02937318", 
      "https://app.dimensions.ai/details/publication/pub.1038620902"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2019-04-11T14:26", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000373_0000000373/records_13071_00000001.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "http://link.springer.com/10.1007/BF02937318"
  }
]
 

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/bf02937318'

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/bf02937318'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf02937318'

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

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf02937318'


 

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

80 TRIPLES      21 PREDICATES      30 URIs      19 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/bf02937318 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author Nd2e0efc3ba894a9fa1881eb7d19c9bda
4 schema:citation https://doi.org/10.1016/0097-3165(72)90102-1
5 https://doi.org/10.1090/s0002-9947-1994-1264149-8
6 https://doi.org/10.2307/2155003
7 schema:datePublished 1996-06
8 schema:datePublishedReg 1996-06-01
9 schema:description LetG be a group. For a natural numberd≥1 letGd denote the subgroup ofG generated by all powersad,a∈G. A. Shalev raised the question if there exists a functionN=N(m, d) such that for anm-generated finite groupG an arbitrary element fromGd can be represented asa1d...aNd,ai∈G. The positive answer to this question would imply that in a finitely generated profinite groupG all power subgroupsGd are closed and that an arbitrary subgroup of finite index inG is closed. In [5,6] the first author proved the existence of such a function for nilpotent groups and for finite solvable groups of bounded Fitting height. Another interpretation of the existence ofN(m, d) is definability of power subgroupsGd (see [10]). In this paper we address the question for finite simple groups. All finite simple groups are known to be 2-generated. Thus, we prove the following: THEOREM:There exists a function N=N(d) such that for an arbitrary finite simple group G either Gd=1 orG={a1d...aNd|ai∈G}. The proof is based on the Classification of finite simple groups and sometimes resorts to a case-by-case analysis.
10 schema:genre research_article
11 schema:inLanguage en
12 schema:isAccessibleForFree false
13 schema:isPartOf Nb06966305b8640b6a0f97443785f320d
14 Nf9ccaf4788d34c43a17fe9a0e182a031
15 sg:journal.1136632
16 schema:name Products of powers in finite simple groups
17 schema:pagination 469-479
18 schema:productId N01f7a48b306e4cdf861a672b758a0b4c
19 N93749fdeed58441391b69a66827fe473
20 Nbb917e0f2d4143afa2b516dc34509823
21 schema:sameAs https://app.dimensions.ai/details/publication/pub.1038620902
22 https://doi.org/10.1007/bf02937318
23 schema:sdDatePublished 2019-04-11T14:26
24 schema:sdLicense https://scigraph.springernature.com/explorer/license/
25 schema:sdPublisher N785c0ccbdaca40f3882eec4ce8b2db51
26 schema:url http://link.springer.com/10.1007/BF02937318
27 sgo:license sg:explorer/license/
28 sgo:sdDataset articles
29 rdf:type schema:ScholarlyArticle
30 N01f7a48b306e4cdf861a672b758a0b4c schema:name doi
31 schema:value 10.1007/bf02937318
32 rdf:type schema:PropertyValue
33 N785c0ccbdaca40f3882eec4ce8b2db51 schema:name Springer Nature - SN SciGraph project
34 rdf:type schema:Organization
35 N93749fdeed58441391b69a66827fe473 schema:name dimensions_id
36 schema:value pub.1038620902
37 rdf:type schema:PropertyValue
38 Nb06966305b8640b6a0f97443785f320d schema:volumeNumber 96
39 rdf:type schema:PublicationVolume
40 Nb30ac578f2664130b4207a602b9c4c23 rdf:first sg:person.01264523514.92
41 rdf:rest rdf:nil
42 Nbb917e0f2d4143afa2b516dc34509823 schema:name readcube_id
43 schema:value b7f8bb3645ff8f823b34e78913131e40c19227c5dd999035fcdc3234936d3afb
44 rdf:type schema:PropertyValue
45 Nd2e0efc3ba894a9fa1881eb7d19c9bda rdf:first sg:person.015261576461.61
46 rdf:rest Nb30ac578f2664130b4207a602b9c4c23
47 Nf9ccaf4788d34c43a17fe9a0e182a031 schema:issueNumber 2
48 rdf:type schema:PublicationIssue
49 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
50 schema:name Mathematical Sciences
51 rdf:type schema:DefinedTerm
52 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
53 schema:name Pure Mathematics
54 rdf:type schema:DefinedTerm
55 sg:journal.1136632 schema:issn 0021-2172
56 1565-8511
57 schema:name Israel Journal of Mathematics
58 rdf:type schema:Periodical
59 sg:person.01264523514.92 schema:affiliation https://www.grid.ac/institutes/grid.170205.1
60 schema:familyName Zelmanov
61 schema:givenName E.
62 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.01264523514.92
63 rdf:type schema:Person
64 sg:person.015261576461.61 schema:affiliation https://www.grid.ac/institutes/grid.10863.3c
65 schema:familyName Martinez
66 schema:givenName C.
67 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015261576461.61
68 rdf:type schema:Person
69 https://doi.org/10.1016/0097-3165(72)90102-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1017584976
70 rdf:type schema:CreativeWork
71 https://doi.org/10.1090/s0002-9947-1994-1264149-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1006073338
72 rdf:type schema:CreativeWork
73 https://doi.org/10.2307/2155003 schema:sameAs https://app.dimensions.ai/details/publication/pub.1069793668
74 rdf:type schema:CreativeWork
75 https://www.grid.ac/institutes/grid.10863.3c schema:alternateName University of Oviedo
76 schema:name Departamento de Matemáticas, Universidad de Oviedo, 33.007, Oviedo, Spain
77 rdf:type schema:Organization
78 https://www.grid.ac/institutes/grid.170205.1 schema:alternateName University of Chicago
79 schema:name Department of Mathematics, University of Chicago, 60637, Chicago, IL, USA
80 rdf:type schema:Organization
 




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


...