×

Random generation of simple groups by two conjugate elements. (English) Zbl 0904.20020

Let \(G\) denote a finite simple group. Let \(P_2(G)\) denote the probability that \(2\) randomly chosen elements of \(G\) generate \(G\). A consequence of the classification of finite simple groups is that \(P_2(G)>0\) for all such \(G\). A recent result due to a combination of the results of J. D. Dixon [Math. Z. 110, 199-205 (1969; Zbl 0176.29901)], W. M. Kantor, A. Lubotzky [Geom. Dedicata 36, No. 1, 67-87 (1990; Zbl 0718.20011)]and M. W. Liebeck, A. Shalev [ibid. 56, No. 1, 103-113 (1995; Zbl 0836.20068)]shows that \(P_2(G)\to 1\) as \(| G|\to\infty\).
In this note, the author considers a variation of the problem suggested by G. Robinson. Let \(G\) be a finite simple group. Let \(P_c(G)\) denote the probability that if \(x\) and \(y\) are chosen randomly, then \(G=\langle x,y\rangle\). Note that \(P_c(G)\leq P_2(G)\). The author shows that for many families of simple groups, \(P_c(G)\to 1\) as \(| G|\to\infty\).
The main idea of the proof is just to compute the probability that \(x\) and \(x^y\) are contained in a maximal subgroup in a given conjugacy class. There are two key ingredients in the proof. The first is to observe that often, one can show that most elements cannot be contained in many of the classes of maximal subgroups. The second is to use fixed point ratios of elements in permutation groups.
We note that in subsequent work of the author and others, the same result has been shown for all finite simple groups (using the classification of finite simple groups). See R. Guralnick, M. Liebeck, J. Saxl and A. Shalev [Random generation of finite simple groups, preprint].

MSC:

20F05 Generators, relations, and presentations of groups
20P05 Probabilistic methods in group theory
20D05 Finite simple groups and their classification
20E28 Maximal subgroups
20E07 Subgroup theorems; subgroup growth
PDFBibTeX XMLCite
Full Text: DOI