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18
Presentations of finite simple groups: a quantitative approach
"... There is a constant C0 such that all nonabelian finite simple groups of rank n over Fq, with the possible exception of the Ree groups 2G2(32e+1), have presentations with at most C0 generators and relations and total length at most C0(log n + log q). As a corollary, we deduce a conjecture of Holt: th ..."
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There is a constant C0 such that all nonabelian finite simple groups of rank n over Fq, with the possible exception of the Ree groups 2G2(32e+1), have presentations with at most C0 generators and relations and total length at most C0(log n + log q). As a corollary, we deduce a conjecture of Holt: there is a constant C such that dim H2 (G, M) ≤ C dim M for every finite simple group G, every prime p and every irreducible FpGmodule M.
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
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Generalizing boolean satisfiability II: Theory
, 2004
"... This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contai ..."
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This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying zap, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the DavisPutnamLogemannLoveland inference procedure to this broader setting, and show that earlier computational improvements are either subsumed or left intact by the new method. The third paper in this series discusses zap’s implementation and presents experimental performance results. 1.
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 ..."
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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) ..."
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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 motivated by 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...
Counting overlattices in automorphism groups of trees
"... We give an upper bound for the number uΓ(n) of “overlattices ” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a 2pregular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well. Nous ..."
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We give an upper bound for the number uΓ(n) of “overlattices ” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a 2pregular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well. Nous donnons une borne supérieure pour le nombre uΓ(n) de “surréseaux ” contenant un réseau fixé d’indice n dans le groupe d’automorphismes d’un arbre. Dans le cas d’un arbre 2prégulier T, et d’un réseau Γ tel que Γ\T soit une boucle, nous obtenons aussi une minoration du comportement asymptotique. Introduction. Given a connected semisimple Lie group G, the KazhdanMargulis lemma says that there exists a positive lower bound for the covolume of cocompact lattices in G. This is no longer true when G is the automorphism group of a locally finite tree. Bass and Kulkarni (for cocompact lattices, see [BK]) and Carbone and Rosenberg (for arbitrary lattices in uniform trees, see [CR]) even constructed examples of increasing sequences of lattices (Γi)i∈N in Aut(T) whose
COUNTING CONGRUENCE SUBGROUPS
"... Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence ..."
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Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence
MONOTONE HURWITZ NUMBERS AND THE HCIZ INTEGRAL II
"... Abstract. Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking a ..."
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Abstract. Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking analogues of known results for the classical Hurwtiz numbers. These include explicit formulas for monotone Hurwitz numbers in genus 0 and 1, for all partitions, and an explicit rational form for the generating series in arbitrary genus. This rational form implies that, up to an explicit combinatorial scaling,
Counting overlattices for polyhedral complexes, submitted
"... Abstract. We investigate the asymptotics of the number of “overlattices” of a cocompact lattice Γ in Aut(X), where X is a locally finite polyhedral complex. We use complexes of groups to prove an upper bound for general X, and a lower bound for certain rightangled buildings. 1. ..."
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Abstract. We investigate the asymptotics of the number of “overlattices” of a cocompact lattice Γ in Aut(X), where X is a locally finite polyhedral complex. We use complexes of groups to prove an upper bound for general X, and a lower bound for certain rightangled buildings. 1.