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Random Generation of Simple Groups by Two Conjugate Elements

Random Generation of Simple Groups by Two Conjugate Elements Abstract Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as ∣G∣→∞. Geoff Robinson asked whether the conclusion still holds if we require further that x, y are conjugate in G. In this note we study the probability Pc(G) that 〈x, xy〉 = G, where x, y∈G are chosen at random (with uniform distribution on G×G). We shall show that Pc(G)→1 as ∣G∣→∞ if G is an alternating group, or a projective special linear group, or a classical group of bounded dimension. In fact, some (but not all) of the exceptional groups of Lie type will also be dealt with. 1991 Mathematics Subject Classification 20E18, 20E07. © London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Random Generation of Simple Groups by Two Conjugate Elements

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References (10)

Publisher
Oxford University Press
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S002460939700338X
Publisher site
See Article on Publisher Site

Abstract

Abstract Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as ∣G∣→∞. Geoff Robinson asked whether the conclusion still holds if we require further that x, y are conjugate in G. In this note we study the probability Pc(G) that 〈x, xy〉 = G, where x, y∈G are chosen at random (with uniform distribution on G×G). We shall show that Pc(G)→1 as ∣G∣→∞ if G is an alternating group, or a projective special linear group, or a classical group of bounded dimension. In fact, some (but not all) of the exceptional groups of Lie type will also be dealt with. 1991 Mathematics Subject Classification 20E18, 20E07. © London Mathematical Society

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Sep 1, 1997

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