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The homotopy theory of fusion systems
"... The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group con ..."
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Cited by 48 (10 self)
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The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like pcompleted classifying spaces of finite groups. These spaces occur as the “classifying spaces ” of certain algebraic objects, which we call plocal finite groups. A plocal finite group consists, roughly speaking, of a finite pgroup S and fusion data on subgroups of S, encoded in a way explained below. Our starting point is our earlier paper [BLO] on pcompleted classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig [Pu], [Pu2] of systems of fusion among subgroups of a given pgroup. The pcompletion of a space X is a space X ∧ p which isolates the properties of X at the prime p, and more precisely the properties which determine its mod p cohomology. For example, a map of spaces X f −− → Y induces a homotopy equivalence
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 ..."
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Cited by 40 (11 self)
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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 ..."
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Cited by 31 (8 self)
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. 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...
Hyperelliptic jacobians without complex multiplication
 Math. Res. Letters
"... The aim of this note is to prove that in positive characteristic p ̸ = 2 the jacobian J(C) = J(Cf) of a hyperelliptic curve C = Cf: y 2 = f(x) has only trivial endomorphisms over an algebraic closure of the ground field ..."
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Cited by 25 (3 self)
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The aim of this note is to prove that in positive characteristic p ̸ = 2 the jacobian J(C) = J(Cf) of a hyperelliptic curve C = Cf: y 2 = f(x) has only trivial endomorphisms over an algebraic closure of the ground field
Probabilistic Recognition of Orthogonal and Symplectic Groups: Corrections
 In Groups and Computation III
"... We propose a simple one sided MonteCarlo algorithm to distinguish, to any given degree of certainty, the symplectic group Cn (q) = PSp 2n (q) from the orthogonal group Bn (q) =# 2n+1 (q) where q > 3 is odd and n and q are given. The algorithm does not use an order oracle and works in polynomial ..."
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Cited by 22 (5 self)
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We propose a simple one sided MonteCarlo algorithm to distinguish, to any given degree of certainty, the symplectic group Cn (q) = PSp 2n (q) from the orthogonal group Bn (q) =# 2n+1 (q) where q > 3 is odd and n and q are given. The algorithm does not use an order oracle and works in polynomial, of n log q, time. This paper corrects an error in the previously published version of the algorithm [1]. 1991 Mathematics Subject Classification: primary 51F15; secondary 20G40. 1.
Periodic complexes and group actions
, 2001
"... In this paper we show that the cohomology of a connected CW–complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m; we ..."
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Cited by 18 (0 self)
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In this paper we show that the cohomology of a connected CW–complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m; we construct non–standard free actions of rank two simple groups on finite complexes Y ≃ S n × S m; and we prove that a finite p–group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p)³.
A brief history of the classification of finite simple groups
 BAMS
"... Abstract. We present some highlights of the 110year project to classify the finite simple groups. ..."
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Cited by 16 (0 self)
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Abstract. We present some highlights of the 110year project to classify the finite simple groups.