In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 ̸ = x ∈ G, the probability is greater than 1/10 that G =〈x,y 〉 for a random y ∈ C. Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds. We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x1, x2 are nontrivial elements of G, then there exists y ∈ C such that G =〈x1,y〉=〈x2,y〉. Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x1, x2, x3 are nontrivial elements of G, then there exists y ∈ C such that G =〈x1,y〉= 〈x2,y〉=〈x3,y〉. We also prove analogous but weaker results for almost simple groups.

. The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating k-tuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...

The diameter of a group G with respect to a set S of generators is the maximum over g 2 G of the length of the shortest word in S [ S 1 representing g. This concept arises in the contexts of efficient communication networks and Rubik's cube type puzzles. "Best" generators (giving minimum diameter while keeping the number of generators limited) are pertinent to networks, "worst" and "average" generators seem a more adequate model for puzzles. We survey a substantial body of recent work by the authors on these subjects. Regarding the "best" case, we show that while the structure of the group is essentially irrelevant if |S| is allowed to exceed (log |G|) 1+c (c > 0), it plays a heavy role when jSj = O(1). In particular, every nonabelian nite simple group has a set of 7 generators giving logarithmic diameter. This cannot happen for groups with an abelian subgroup of bounded index. { Regarding the worst case, we are concerned primarily with permutation groups of degree n and obtain a tight exp((n ln n) 1=2 (1 + o(1))) upper bound. In the average case, the upper bound improves to exp((ln n) 2 (1 + o(1))). As a rst step toward extending this result to simple groups other than An , we establish that almost every pair of elements of a classical simple group G generates G, a result previously proved by J. Dixon for An . In the limited space of this article, we try to illuminate some of the basic underlying techniques.

Summary. Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time ” of the chain, that is, the time at which the chain gives a good approximation of the limit distribution. A remarkable phenomenon known as the cut-off phenomenon asserts that this often happens abruptly so that it really makes sense to talk about “the mixing time”. Random walks on finite groups generalize card shuffling models by replacing the symmetric group by other finite groups. One then would like to understand how the structure of a particular class of groups relates to the mixing time of natural random walks on those groups. It turns out that this is an extremely rich problem which is very far to be understood. Techniques from a great

In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the probabilistic nature of the problem is apparent from its very formulation (see [KL], [GKS], [LiSh1]); but in other cases the use of probability,

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The study of asymptotics of random permutations was initiated by Erdos and Tunto. in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more grouptheoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions). 1. Permutations lying in a transitive subgroup Sn and An denote the symmetric and alternating groups on the set X = {I,.... n}. A subgroup G of S " is transitive if, for all i, j E X, there exists g E G with ig ~ j. In a preliminary version of this paper, we asked the following question: Question 1.1. Is it true that,/or almost all permutations g E Sn. the only transitive subgroups containing g are Sn and (possihly) An? Here, of course, 'almost all g E S " have property P ' means 'the proportion of elements of S " not having property P tends to 0 as n--> ex'. An affirmative answer to this question was given by Luczak and Pyber, in [15]. We will

. We establish a connection between the expansion coefficient of the product replacement graph \Gamma k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs \Gamma k \Gamma PSL(2; p) \Delta form an expander family, under assumption that all Cayley graphs of PSL(2; p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [LP,P3].

) IGOR PAK , SERGEY BRATUS y 1 Introduction Let G be a finite group. A sequence of k group elements (g1 ; : : : ; gk ) is called a generating k-tuple of G if the elements generate G (we write hg1 ; : : : ; gk i = G). Let Nk (G) be the set of all generating k-tuples of G, and let Nk (G) = jNk (G)j. We consider two related problems on generating k-tuples. Given G and k ? 0, 1) Determine Nk (G) 2) Generate random element of Nk (G), each with probability 1=Nk (G) The problem of determining the structure of Nk (G) is of interest in several contexts. The counting problem goes back to Philip Hall, who expressed Nk (G) as a Mobius type summation of Nk (H) over all maximal subgroups H ae G (see [23]). Recently the counting problem has been studied for large simple groups where remarkable progress has been made (see [25, 27]). In this paper we analyze Nk for solvable groups and products of simple groups. The sampling problem, while often used in theory as a tool for approximate counting...

are chosen at random from the symmetric group Sn of degree n. What is the probability that they will generate Sn? " Actually, Netto conjectured last century that almost all pairs of elements from Sn will generate Sn or An. Dixon showed that this is true