2013
DOI: 10.1007/s11856-013-0034-7
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The probability of generating a finite simple group

Abstract: We study the probability of generating a finite simple group, together with its generalisation P G,soc G (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ soc G. We prove that P G,soc G (2) 53/90, with equality if and only if G is A 6 or S 6 , and establish a similar result for P G,soc G (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel nikov on profinite groups, as well as to a conjecture … Show more

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Cited by 22 publications
(18 citation statements)
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“…However Table 1 of [15] reveals that the generating graphs of S 8 and G are different when |G| = 8! and T ∼ = PSL 3 (4).…”
Section: Symmetric Groupsmentioning
confidence: 99%
See 2 more Smart Citations
“…However Table 1 of [15] reveals that the generating graphs of S 8 and G are different when |G| = 8! and T ∼ = PSL 3 (4).…”
Section: Symmetric Groupsmentioning
confidence: 99%
“…Since r < (log 2 n)/(log 2 |T |) < (log 2 n)/5, there are fewer than (log 2 n)/5 choices for r. By [10, Theorem 6.1], for each r there are at most two choices for T . Finally, by[16, Lemma 2.1], if m = 2, then there are at most (6/7) log 2 |T | ≤ (6/7) log 2 n choices for x (any choice of x projecting to a fixed element of Out(T ) gives rise to the same group G), and one if m = 1. Thus there are fewer than (12/35)(log 2 n) 2 + (2/5) log 2 n possibilities for G. In fact, we can divide this bound by 2 by using the fact that if there exist two nonisomorphic simple groups of order |T |, then |T | ≥ 20 160.…”
mentioning
confidence: 99%
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“…We will refer to it as Corollary 1 in [11]. We also use recent estimates obtained by Menezes, Quick and Roney-Dougal [14].…”
Section: Examplementioning
confidence: 99%
“…The proof of part (v)(a) and (v)(b) of Theorem 1.1 are obvious if one uses an observation (cf. Lemma 5 of [13]) that two elements generate a product of finite simple groups [14]) on the number h in a direct product H h (H is a finite simple group) for which it is possible to be generated by 2 elements.…”
Section: Proofmentioning
confidence: 99%