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Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements
 Trans. Amer. Math. Soc
"... Abstract. We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the ..."
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Abstract. We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the BostonShalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third). 1.
Abelian unipotent subgroups of reductive groups
 J. Pure Appl. Algebra
"... ABSTRACT. Let G be a connected reductive group defined over an algebraically closed field k of characteristic p> 0. The purpose of this paper is twofold. First, when p is a good prime, we give a new proof of the “order formula ” of D. Testerman for unipotent elements in G; moreover, we show that ..."
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Cited by 23 (4 self)
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ABSTRACT. Let G be a connected reductive group defined over an algebraically closed field k of characteristic p> 0. The purpose of this paper is twofold. First, when p is a good prime, we give a new proof of the “order formula ” of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the pnilpotence degree of the corresponding nilpotent elements in the Lie algebra g of G. Second, if X is a pnilpotent element (an element of pnilpotence degree 1) in g, we show that G always has a representation V for which the exponential homomorphism Ga → GL(V) determined by X factors through the action of G. This property permits a simplification of the description given by Suslin, Friedlander, and Bendel of the (even) cohomology ring for the higher Frobenius kernels Gd, d ≥ 2. 1.
On the centralizer of the sum of commuting nilpotent elements
 J. Pure Appl. Algebra206
"... Abstract. Let X and Y be commuting nilpotent Kendomorphisms of a vector space V, where K is a field of characteristic p ≥ 0. If F = K(t) is the field of rational functions on the projective line P 1 /K, consider the K(t)endomorphism A = X +tY of V. If p = 0, or if A p−1 = 0, we show here that X an ..."
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Cited by 5 (0 self)
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Abstract. Let X and Y be commuting nilpotent Kendomorphisms of a vector space V, where K is a field of characteristic p ≥ 0. If F = K(t) is the field of rational functions on the projective line P 1 /K, consider the K(t)endomorphism A = X +tY of V. If p = 0, or if A p−1 = 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V). For all geometric points (a: b) of a suitable open subset of P 1, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX + bY. This answers a question of J. Pevtsova. Let G be a connected and reductive algebraic group defined over an arbitrary field K of characteristic p ≥ 0. Write g = Lie(G), and consider the extension field F = K(t) with t transcendental over K. For convenience, we fix an algebraically closed field k containing both K and t. If X,Y ∈ g(K) are nilpotent and [X,Y] = 0, then A = X + tY ∈ g(F) is again nilpotent. Write C for the centralizer of A in G, and write RuC for the unipotent
Primitive permutation groups of bounded orbital diameter
, 2008
"... We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A k ..."
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We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specific cases, there is such a diameter bound. This is proved using recent results on the model theory of pseudofinite fields and difference fields. 1
Infiniteness of double coset collections in algebraic groups’, submitted
"... Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is X\G/P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This pa ..."
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Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is X\G/P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those X which are spherical. Finally, excluding a case in F4, we show that if X\G/P is finite then X is spherical or the Levi factor of P is spherical. This implies that it is rare for X\G/P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.
Permutation Groups in OMinimal Structures
"... Introduction In this paper we develop a structure theory for transitive permutation groups definable in ominimal structures. We fix an ominimal structure M, a group G definable in M, and a set\Omega and an action of G on\Omega definable in M, and talk of the permutation group (G; \Omega\Gamma ..."
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Introduction In this paper we develop a structure theory for transitive permutation groups definable in ominimal structures. We fix an ominimal structure M, a group G definable in M, and a set\Omega and an action of G on\Omega definable in M, and talk of the permutation group (G; \Omega\Gamma/ Often, we are concerned with definably primitive permutation groups (G; \Omega\Gamma/ this means that there is no proper nontrivial definable Ginvariant equivalence relation on \Omega\Gamma so it is equivalent to a point stabiliser G ff being a maximal definable subgroup of G. Of course, since any group definable in an ominimal structure has the descending chain condition on definable subgroups [20] we expect many questions on definable transitive permutation groups to reduce to questi
A classification of certain finite double coset collections in the classical groups
 Bull. London Math. Soc
"... Let G be a classical algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using geometric arguments about the action of X on subspaces of the natural module for G. Infiniteness is proven using a dimension cr ..."
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Let G be a classical algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using geometric arguments about the action of X on subspaces of the natural module for G. Infiniteness is proven using a dimension criterion which involves root systems. 1 Statement of results Let G be a classical algebraic group defined over an algebraically closed field, let X be a maximal rank reductive subgroup, and let P be a parabolic subgroup. The property of finiteness for X\G/P is preserved under taking isogenies, quotients by the center of G, connected components and conjugates (see Lemma 2.2 for a precise statement). Thus, if desired, we can specify only the Lie type of G. Similarly, we can specify only the conjugacy class of X and P; thus we usually give the Lie type of X and describe P by crossing off nodes from the Dynkin diagram for G. For the purpose of classifying finiteness, it suffices to consider only those X which are defined over Z. A subgroup X is spherical if X\G/B is finite for some (or, equivalently, for each) Borel subgroup B. For each classical group we list in Table 1 those maximal rank reductive spherical subgroups which are defined over Z. We first describe the notation which is used for the list, and for the rest of the paper, and then describe how the list is obtained. We write X = AnAmT1 if X is a group of Lie type An + Am which has a 1dimensional central torus, orbits
Topics in the Theory of Algebraic Groups
"... This article is a collection of notes from a series of talks given at the Bernoulli center. The attendees ranged from people who have never studied algebraic groups to experts. Consequently the series began with two introductory talks on the structure of algebraic groups, supplemented by two lecture ..."
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This article is a collection of notes from a series of talks given at the Bernoulli center. The attendees ranged from people who have never studied algebraic groups to experts. Consequently the series began with two introductory talks on the structure of algebraic groups, supplemented by two lectures of Steve Donkin on representation theory. The notes
Affine distancetransitive graphs and . . .
, 2003
"... This paper finishes the classification of primitive affine distancetransitive graphs by dealing with the only case left open, namely where the generalized Fitting subgroup of the stabilizer of a vertex is a finite quasisimple group of classical Lie type defined over the characteristic dividing the ..."
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This paper finishes the classification of primitive affine distancetransitive graphs by dealing with the only case left open, namely where the generalized Fitting subgroup of the stabilizer of a vertex is a finite quasisimple group of classical Lie type defined over the characteristic dividing the number of vertices of the graph. All graphs that are found to occur are known.