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33
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.
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 35 (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
Simple groups, permutation groups, and probability
 In Proceedings of the International Congress of Mathematicians
, 1999
"... 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 ..."
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Cited by 20 (0 self)
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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,
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.
On the subgroup structure of exceptional groups of Lie type
 Trans. Amer. Math. Soc
, 1998
"... Abstract. We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q) ofLie type in the natural characteristic. Our approach is to show that for sufficie ..."
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Cited by 9 (4 self)
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Abstract. We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X(q) ofLie type in the natural characteristic. Our approach is to show that for sufficiently large q (usually q>9 suffices), X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.
Composition factors from the group ring and Artin's theorem on orders of simple groups
 Proc. London Math. Soc
, 1990
"... The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. Th ..."
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Cited by 8 (2 self)
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The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups. 1.
Fixed point ratios in actions of finite exceptional groups of Lie type
 Pacific J. Math
"... Let G be a finite exceptional group of Lie type acting transitively on a set Ω. For x ∈ G, the fixed point ratio of x is the proportion of elements of Ω which are fixed by x. We obtain new bounds for such fixed point ratios. When a pointstabilizer is parabolic we use character theory; and in other c ..."
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Cited by 8 (2 self)
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Let G be a finite exceptional group of Lie type acting transitively on a set Ω. For x ∈ G, the fixed point ratio of x is the proportion of elements of Ω which are fixed by x. We obtain new bounds for such fixed point ratios. When a pointstabilizer is parabolic we use character theory; and in other cases, we use results on an analogous problem for algebraic groups in Lawther, Liebeck & Seitz, 2002. These give dimension bounds on fixed point spaces of elements of exceptional algebraic groups, which we apply by passing to finite groups via a Frobenius morphism. Introduction. If G is a finite group acting transitively on a set Ω, and x ∈ G, we define the fixed point ratio of x to be the proportion of points fixed by x; that is, denoting this quantity by fpr(x, Ω),
Derangements in simple and primitive groups
 2001 Conference on Groups, Combinatorics, and Geometry
"... Abstract. We investigate the proportion of fixed point free permutations (derangements) in finite transitive permutation groups. This article is the first in a series where we prove a conjecture of Shalev that the proportion of such elements is bounded away from zero for a simple finite group. In fa ..."
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Cited by 8 (3 self)
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Abstract. We investigate the proportion of fixed point free permutations (derangements) in finite transitive permutation groups. This article is the first in a series where we prove a conjecture of Shalev that the proportion of such elements is bounded away from zero for a simple finite group. In fact, there are much stronger results. This article focuses on finite Chevalley groups of bounded rank. We also discuss derangements in algebraic groups and in more general primitive groups. These results have applications in questions about probabilistic generation of finite simple groups and maps between varieties over finite fields. 1.
Finding Sylow normalizers in polynomial time
 JOURNAL OF ALGORITHMS
, 1990
"... Given a set r of permutations of an nset, let G be the group of permutations generated by r. If p is any prime, it is known that a Sylow psubgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow psubgrou ..."
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Cited by 7 (4 self)
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Given a set r of permutations of an nset, let G be the group of permutations generated by r. If p is any prime, it is known that a Sylow psubgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow psubgroups of G, all elements conjugating one to the other can be found (as a coset of the normalizer of one of the Sylow psubgroups). Analogous results are obtained in the case of Hall subgroups of solvable groups.
Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable
 J. Amer. Math. Soc
"... The purpose of this paper is to prove the following. Main Theorem. Let D be a finite dimensional division algebra. Then any finite quotient of the multiplicative group D × is solvable. This result is a culmination of research done in the last several years in order ..."
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Cited by 6 (0 self)
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The purpose of this paper is to prove the following. Main Theorem. Let D be a finite dimensional division algebra. Then any finite quotient of the multiplicative group D × is solvable. This result is a culmination of research done in the last several years in order