On the probability of generating a monolithic group

Eloisa Detomi; Andrea Lucchini; Colva Mary Roney-Dougal

doi:10.1016/j.jalgebra.2014.03.010

On the probability of generating a monolithic group

Eloisa Detomi, Andrea Lucchini, Colva Mary Roney-Dougal

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Abstract

A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By P_L,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that P_L,N(k) is controlled by P_Y,S(2), where N≅S^rand Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N . Information about P_L,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.

Original language	English
Pages (from-to)	413-433
Journal	Journal of Algebra
Volume	407
Early online date	18 Apr 2014
DOIs	https://doi.org/10.1016/j.jalgebra.2014.03.010
Publication status	Published - 1 Jun 2014

Keywords

Finite group theory
Random generation of finite groups
Finite simple groups
Crown-based power

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10.1016/j.jalgebra.2014.03.010Licence: CC BY

JoA2014
Copyright 2014, the authors. This is an open access article under the CC-BY license. http://creativecommons.org/licenses/by/3.0/
Final published version, 373 KBLicence: CC BY

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

Colva Roney-Dougal, OBE
- Colva.Roney-Dougalst-andrews.acuk
Person: Academic

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title = "On the probability of generating a monolithic group",

abstract = "A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N . Information about PL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.",

keywords = "Finite group theory , Random generation of finite groups , Finite simple groups , Crown-based power",

author = "Eloisa Detomi and Andrea Lucchini and Roney-Dougal, {Colva Mary}",

note = "This research was supported through EPSRC grant EP/I03582X/1. The APC was paid through RCUK open access block grant funds. ",

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doi = "10.1016/j.jalgebra.2014.03.010",

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T1 - On the probability of generating a monolithic group

AU - Detomi, Eloisa

AU - Lucchini, Andrea

AU - Roney-Dougal, Colva Mary

N1 - This research was supported through EPSRC grant EP/I03582X/1. The APC was paid through RCUK open access block grant funds.

PY - 2014/6/1

Y1 - 2014/6/1

N2 - A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N . Information about PL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.

AB - A group L is primitive monolithic if L has a unique minimal normal subgroup, N , and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L , given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N . Information about PL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.

KW - Finite group theory

KW - Random generation of finite groups

KW - Finite simple groups

KW - Crown-based power

U2 - 10.1016/j.jalgebra.2014.03.010

DO - 10.1016/j.jalgebra.2014.03.010

M3 - Article

SN - 0021-8693

VL - 407

SP - 413

EP - 433

JO - Journal of Algebra

JF - Journal of Algebra

ER -

On the probability of generating a monolithic group

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

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Colva Roney-Dougal, OBE

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