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 of Holt and Stather, follow easily from our results.

On the Generating Graph of a Simple Group

“…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).…”

“…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.…”

“…In [5, pages 239-242], there is a list of simple groups S with |S | < 10 25 . By going through this list, we see that the only groups that can appear as T , the socle of G, are PSL 2 (q) with |T | > 10 6 ; PSL 3 (q) with |T | > 10 12 ; PSU 3 (q) with |T | > 10 12 ; PSL 4 (q) with |T | > 10 16 ; PSU 4 (q) with |T | > 10 16 ; PSp 4 (q) with |T | > 10 16 ; and G 2 (q) with |T | > 10 20 .…”

On the generation of discrete and topological Kac–Moody groups

“…We will refer to it as Corollary 1 in [11]. We also use recent estimates obtained by Menezes, Quick and Roney-Dougal [14].…”

“…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.…”