. 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...

Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, Rogers-Ramanujan type identities, potential theory, and various measures on partitions.

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

The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating k-tuples of the group, and then output a random component. The algorithm was designed by Leedham-Green and Soicher ([31]), and further investigated in [12]. It was found to have an outstanding performance, much better than the the previously known algorithms (see [12, 22, 26]). The algorithm is now included in two major group algebra packages GAP [42] and MAGMA [10]. In spite of the many serious attempts and partial results, (see [6, 14, 15, 21, 22, 32, 39, 40]), the analysis of the algorithm remains difficult at best. For small values of k even graph connectivity becomes a serious obstacle (see [19, 37, 39, 40]). The most general results are due to Diaconis and Saloff--Coste [22], who used a state of the art analytic technique to obtain polynomial bounds in special cases, and (sub)-exponential bounds in general case. The main result of this pape...

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) 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...

. Let G be a finite group, and let ' k (G) be the probability that k random group elements generate G. Denote by #(G) the smallest k such that ' k (G) ? 1=e. In this paper we analyze quantity #(G) for different classes of groups. We prove that #(G) --(G) + 1 when G is nilpotent and --(G) is the minimal number of generators of G. When G is solvable we show that #(G) 3:25 --(G) + 10 7 . We also show that #(G) ! C log log jGj, where G is a direct product of simple nonabelian groups, and C is a universal constant. The work is motivatedby the applications to the "product replacement algorithm" (see [CLMNO,P4]). This algorithm is an important recent innovation, designed to efficiently generate (nearly) uniform random group elements. Recent work by Babai and the author [BaP] showed that the output of the algorithm must have a strong bias in certain cases. The precise probabilistic estimates we obtain here, combined with a note [P3], give positive result, proving that no bias exists for...

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. We prove that the product replacement graph on generating 3-tuples of An is connected for n 11. We employ an efficient heuristic based on [P1] which works significantly faster than brute force. The heuristic works for any group. Our tests were confined to An due to the interest in Wiegold's Conjecture, usually stated in terms of T -systems (see [P2]). Our results confirm Wiegold's Conjecture in some special cases and are related to the recent conjecture of Diaconis and Graham [DG]. The work was motivated by the study of the product replacement algorithm (see [CLMNO,P2]). Introduction Let G be a finite group, and let N k (G) be the set of generating k-tuples (g) = (g 1 ; : : : ; g k ), where hg 1 ; : : : ; g k i = G. Define moves on N k (G) as follows: R \Sigma i;j : (g 1 ; : : : ; g i ; : : : ; g k ) ! (g 1 ; : : : ; g i \Delta g \Sigma1 j ; : : : ; g k ) L \Sigma i;j : (g 1 ; : : : ; g i ; : : : ; g k ) ! (g 1 ; : : : ; g \Sigma1 j \Delta g i ; : : : ; g k ) ) ...

Dimension and randomness in groups acting on rooted trees