Invariable generation and the Chebotarev invariant of a finite group

W. M. Kantor; A. Lubotzky; A. Shalev

doi:10.1016/j.jalgebra.2011.09.022

Invariable generation and the Chebotarev invariant of a finite group

W. M. Kantor^*, A. Lubotzky, A. Shalev

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

38 Scopus citations

Abstract

A subset S of a finite group G invariably generates G if G=〈s^g(s)|s∈S〉 for each choice of g(s)∈G, s∈S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in Kowalski and Zywina (2010) [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.

Original language	English
Pages (from-to)	302-314
Number of pages	13
Journal	Journal of Algebra
Volume	348
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2011.09.022
State	Published - 15 Dec 2011

Bibliographical note

Funding Information:
✩ The authors acknowledge partial support from NSF grant DMS 0753640 (W.M.K.), ERC Advanced Grants 226135 (A.L.) and 247034 (A.S.), and ISF grant 754/08 (A.L. and A.S.). The first author is grateful for the warm hospitality of the Hebrew University while this paper was being written. * Corresponding author. E-mail addresses: [email protected] (W.M. Kantor), [email protected] (A. Lubotzky), [email protected] (A. Shalev).

Keywords

Chebotarev invariant
Invariable generation
Simple group

Access to Document

10.1016/j.jalgebra.2011.09.022

Cite this

@article{7b122901689641b8afab4f421c41a992,

title = "Invariable generation and the Chebotarev invariant of a finite group",

abstract = "A subset S of a finite group G invariably generates G if G=〈sg(s)|s∈S〉 for each choice of g(s)∈G, s∈S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in Kowalski and Zywina (2010) [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.",

keywords = "Chebotarev invariant, Invariable generation, Simple group",

author = "Kantor, {W. M.} and A. Lubotzky and A. Shalev",

note = "Funding Information: ✩ The authors acknowledge partial support from NSF grant DMS 0753640 (W.M.K.), ERC Advanced Grants 226135 (A.L.) and 247034 (A.S.), and ISF grant 754/08 (A.L. and A.S.). The first author is grateful for the warm hospitality of the Hebrew University while this paper was being written. * Corresponding author. E-mail addresses: [email protected] (W.M. Kantor), [email protected] (A. Lubotzky), [email protected] (A. Shalev).",

year = "2011",

month = dec,

day = "15",

doi = "10.1016/j.jalgebra.2011.09.022",

language = "אנגלית",

volume = "348",

pages = "302--314",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

number = "1",

}

TY - JOUR

T1 - Invariable generation and the Chebotarev invariant of a finite group

AU - Kantor, W. M.

AU - Lubotzky, A.

AU - Shalev, A.

N1 - Funding Information: ✩ The authors acknowledge partial support from NSF grant DMS 0753640 (W.M.K.), ERC Advanced Grants 226135 (A.L.) and 247034 (A.S.), and ISF grant 754/08 (A.L. and A.S.). The first author is grateful for the warm hospitality of the Hebrew University while this paper was being written. * Corresponding author. E-mail addresses: [email protected] (W.M. Kantor), [email protected] (A. Lubotzky), [email protected] (A. Shalev).

PY - 2011/12/15

Y1 - 2011/12/15

N2 - A subset S of a finite group G invariably generates G if G=〈sg(s)|s∈S〉 for each choice of g(s)∈G, s∈S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in Kowalski and Zywina (2010) [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.

AB - A subset S of a finite group G invariably generates G if G=〈sg(s)|s∈S〉 for each choice of g(s)∈G, s∈S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response to a question in Kowalski and Zywina (2010) [KZ] we also bound the size of a randomly chosen set of elements of G that is likely to generate G invariably. Along the way we prove that every finite simple group is invariably generated by two elements.

KW - Chebotarev invariant

KW - Invariable generation

KW - Simple group

UR - http://www.scopus.com/inward/record.url?scp=80755140651&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2011.09.022

DO - 10.1016/j.jalgebra.2011.09.022

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:80755140651

SN - 0021-8693

VL - 348

SP - 302

EP - 314

JO - Journal of Algebra

JF - Journal of Algebra

IS - 1

ER -

Invariable generation and the Chebotarev invariant of a finite group

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this