Generation and random generation: From simple groups to maximal subgroups

Timothy C. Burness; Martin W. Liebeck; Aner Shalev

doi:10.1016/j.aim.2013.07.009

Generation and random generation: From simple groups to maximal subgroups

Timothy C. Burness^*
, Martin W. Liebeck
, Aner Shalev

^*Corresponding author for this work

Einstein Institute of Mathematics

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

11 Scopus citations

Abstract

Let G be a finite group and let d( G) be the minimal number of generators for G. It is well known that d( G) = 2 for all (non-abelian) finite simple groups. We prove that d( H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.We then apply our results to the study of permutation groups. In particular we show that if G is a finite primitive permutation group with point stabilizer H, then d( G) - 1 ≤ d( H) ≤ d( G) + 4.

Original language	English
Pages (from-to)	59-95
Number of pages	37
Journal	Advances in Mathematics
Volume	248
DOIs	https://doi.org/10.1016/j.aim.2013.07.009
State	Published - 5 Nov 2013

Bibliographical note

Funding Information:
The first author acknowledges the support of EPSRC grant EP/I019545/1 . The second and third authors acknowledge the support of EPSRC grant EP/H018891/1 . The third author also acknowledges the support of an Advanced ERC Grant, an Israel Science Foundation Grant, and the Miriam and Julius Vinik Chair in Mathematics which he holds.

Keywords

Finite simple groups
Maximal subgroups
Minimal generation
Primitive permutation groups

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10.1016/j.aim.2013.07.009

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title = "Generation and random generation: From simple groups to maximal subgroups",

abstract = "Let G be a finite group and let d( G) be the minimal number of generators for G. It is well known that d( G) = 2 for all (non-abelian) finite simple groups. We prove that d( H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.We then apply our results to the study of permutation groups. In particular we show that if G is a finite primitive permutation group with point stabilizer H, then d( G) - 1 ≤ d( H) ≤ d( G) + 4.",

keywords = "Finite simple groups, Maximal subgroups, Minimal generation, Primitive permutation groups",

author = "Burness, \{Timothy C.\} and Liebeck, \{Martin W.\} and Aner Shalev",

note = "Funding Information: The first author acknowledges the support of EPSRC grant EP/I019545/1 . The second and third authors acknowledge the support of EPSRC grant EP/H018891/1 . The third author also acknowledges the support of an Advanced ERC Grant, an Israel Science Foundation Grant, and the Miriam and Julius Vinik Chair in Mathematics which he holds.",

year = "2013",

month = nov,

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doi = "10.1016/j.aim.2013.07.009",

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AU - Burness, Timothy C.

AU - Liebeck, Martin W.

AU - Shalev, Aner

N1 - Funding Information: The first author acknowledges the support of EPSRC grant EP/I019545/1 . The second and third authors acknowledge the support of EPSRC grant EP/H018891/1 . The third author also acknowledges the support of an Advanced ERC Grant, an Israel Science Foundation Grant, and the Miriam and Julius Vinik Chair in Mathematics which he holds.

PY - 2013/11/5

Y1 - 2013/11/5

N2 - Let G be a finite group and let d( G) be the minimal number of generators for G. It is well known that d( G) = 2 for all (non-abelian) finite simple groups. We prove that d( H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.We then apply our results to the study of permutation groups. In particular we show that if G is a finite primitive permutation group with point stabilizer H, then d( G) - 1 ≤ d( H) ≤ d( G) + 4.

AB - Let G be a finite group and let d( G) be the minimal number of generators for G. It is well known that d( G) = 2 for all (non-abelian) finite simple groups. We prove that d( H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.We then apply our results to the study of permutation groups. In particular we show that if G is a finite primitive permutation group with point stabilizer H, then d( G) - 1 ≤ d( H) ≤ d( G) + 4.

KW - Finite simple groups

KW - Maximal subgroups

KW - Minimal generation

KW - Primitive permutation groups

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AN - SCOPUS:84882936084

SN - 0001-8708

VL - 248

SP - 59

EP - 95

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Generation and random generation: From simple groups to maximal subgroups

Abstract

Bibliographical note

Keywords

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