Results 1  10
of
15
Probabilistic generation of finite simple groups, II
, 2008
"... 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 ..."
Abstract

Cited by 40 (11 self)
 Add to MetaCart
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.
What Do We Know About The Product Replacement Algorithm?
 in: Groups ann Computation III
, 2000
"... . 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 ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an exten ..."
Abstract

Cited by 30 (7 self)
 Add to MetaCart
. 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 ktuples 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...
Random matrix theory over finite fields
 Bull. Amer. Math. Soc. (N.S
"... 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 sym ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
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, RogersRamanujan type identities, potential theory, and various measures on partitions.
Random Walks On Finite Groups With Few Random Generators
 Electr. J. Prob
, 1999
"... . Let G be a finite group. Choose a set S of size k uniformly from G and consider a lazy random walk on the corresponding Cayley graph. We show that for almost all choices of S given k = 2 a log 2 jGj, a ? 1, this walk mixes in under m = 2a log a a\Gamma1 log jGj steps. A similar result was obtai ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
. Let G be a finite group. Choose a set S of size k uniformly from G and consider a lazy random walk on the corresponding Cayley graph. We show that for almost all choices of S given k = 2 a log 2 jGj, a ? 1, this walk mixes in under m = 2a log a a\Gamma1 log jGj steps. A similar result was obtained earlier by Alon and Roichman (see [AR]), Dou and Hildebrand (see [DH]) using a different techniques. We also prove that when sets are of size k = log 2 jGj+O(log log jGj), m = O(log 3 jGj) steps suffice for mixing of the corresponding symmetric lazy random walk. Finally, when G is abelian we obtain better bounds in both cases. A.M.S. Classification. 60C05,60J15. Key words and phrases. Random random walks on groups, random subproducts, probabilistic method, separation distance. Submitted to EJP on June 5, 1998. Final version accepted on November 11, 1998. Typeset by A M ST E X 2 IGOR PAK Introduction In the past few years there has been a significant progress in analysis of rando...
Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
Expansion of Product Replacement Graphs
 Combinatorica
, 2001
"... . 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 ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
. 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].
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
Maximal subgroups in finite and profinite groups
 Trans. Amer. Math. Soc
, 1996
"... Abstract. We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
Abstract. We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203–220, we then prove that a finite group G has at most G  c maximal soluble subgroups, and show that this result is rather useful in various enumeration problems. 1.
On Probability Of Generating A Finite Group
, 1999
"... . 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 mi ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
. 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...