The probability of generating a finite simple group

Nina Emma Menezes; Martyn Quick; Colva Mary Roney-Dougal

doi:10.1007/s11856-013-0034-7

The probability of generating a finite simple group

Nina Emma Menezes, Martyn Quick, Colva Mary Roney-Dougal

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18 Citations (Scopus)

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Abstract

We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.

Original language	English
Pages (from-to)	371-392
Number of pages	22
Journal	Israel Journal of Mathematics
Volume	198
Issue number	1
Early online date	1 Jul 2013
DOIs	https://doi.org/10.1007/s11856-013-0034-7
Publication status	Published - Nov 2013

Keywords

Probability
Finite simple group
Direct products
Profinite groups

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10.1007/s11856-013-0034-7

Probsimple
© 2013 The Hebrew University Magnes Press, Jerusalem. Published by Springer. The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-013-0034-7
Accepted author manuscript, 283 KB

Solving word problems: Solving word problems via generalisations of small cancellation
Roney-Dougal, C. (PI) & Neunhoeffer, M. (CoI)
EPSRC
1/10/11 → 30/09/14
Project: Standard
Automata Languages Decidability: Automata, Languages, Decidability in Algebra
Ruskuc, N. (PI) & Quick, M. (CoI)
EPSRC
1/03/10 → 31/05/14
Project: Standard

Cite this

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title = "The probability of generating a finite simple group",

abstract = "We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel{\textquoteright}nikov on profinite groups, follow easily from our results.",

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AB - We study the probability of generating a finite simple group, together with its generalisation PG,socG(d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ socG. We prove that PG,socG(2) ⩾ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for PG,socG(3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.

KW - Probability

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KW - Direct products

KW - Profinite groups

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The probability of generating a finite simple group

Abstract

Keywords

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Solving word problems: Solving word problems via generalisations of small cancellation

Automata Languages Decidability: Automata, Languages, Decidability in Algebra

Cite this