Premium
Random Generation of Simple Groups by Two Conjugate Elements
Author(s) -
Shalev Aner
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s002460939700338x
Subject(s) - mathematics , simple (philosophy) , conjugate , simple random sample , pure mathematics , mathematical analysis , demography , epistemology , philosophy , population , sociology
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 P c ( G ) that 〈 x , x y 〉 = G , where x , y ∈ G are chosen at random (with uniform distribution on G × G ). We shall show that P c ( 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.