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86
Finite permutation groups and finite simple groups
 Bull. London Math. Soc
, 1981
"... In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been ..."
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Cited by 130 (4 self)
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In the past two decades, there have been farreaching developments in the problem of determining all finite nonabelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of
Arithmetic groups of higher Qrank cannot act on 1manifolds
 Proc. Amer. Math. Soc
, 1994
"... Abstract. Let T be a subgroup of finite index in SLn(Z) with n> 3. We show that every continuous action of T on the circle 51 or on the real line R factors through an action of a finite quotient of T. This follows from the algebraic fact that central extensions of T are not right orderable. (In p ..."
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Cited by 40 (3 self)
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Abstract. Let T be a subgroup of finite index in SLn(Z) with n> 3. We show that every continuous action of T on the circle 51 or on the real line R factors through an action of a finite quotient of T. This follows from the algebraic fact that central extensions of T are not right orderable. (In particular, T is not right orderable.) More generally, the same results hold if T is any arithmetic subgroup of any simple algebraic group G over Q, with Qrank(G)> 2. 1.
The congruence subgroup problem
, 2003
"... This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the nonspecialists and avoids technical details. ..."
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Cited by 33 (1 self)
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This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the nonspecialists and avoids technical details.
On the notion of geometry over F1
"... We refine the notion of variety over the “field with one element” developed by C. Soulé by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. T its. We then solve an open question of C. So ..."
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Cited by 27 (6 self)
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We refine the notion of variety over the “field with one element” developed by C. Soulé by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. T its. We then solve an open question of C. Soulé by proving, using results of J. T its and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above
REDUCED STANDARD MODULES AND COHOMOLOGY
"... Abstract. First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles’s famous paper [42]. Internal to group theory, 1cohomology plays a role in the general theory of maximal subgroups of finite gro ..."
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Cited by 21 (17 self)
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Abstract. First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles’s famous paper [42]. Internal to group theory, 1cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed in AschbacherScott [5]. One can easily pass to the case where the group acts faithfully, and the underlying module is absolutely irreducible. In this case, R. Guralnick [23] conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1cohomology H1 gen(G, L): = lim q→ ∞ H1 (G(q), L) (see [18]) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L, is bounded by a constant depending only on the root system. This result emerges here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules ∆red (λ), ∇red(λ), indexed by dominant weights λ, for a reductive group G. The modules ∆red (λ) and ∇red(λ) arise naturally from
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 19 (0 self)
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Abstract. We present some highlights of the 110year project to classify the finite simple groups.
Computing in groups of Lie type
 Math. Comp
, 2001
"... Abstract. We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear r ..."
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Cited by 15 (5 self)
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Abstract. We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction. 1.
LowDimensional Lattices IV: The Mass Formula
 Proc. Royal Soc. London, A
, 1988
"... The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, ..."
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Cited by 15 (1 self)
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The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, unlike earlier ones, is visibly invariant under scale changes and dualizing. We use the formula to check the enumeration of lattices of determinant d 25 given in the first paper in this series. We also give tables of the "standard mass", the Lseries S (n / m)m  s (m odd), and genera of lattices of determinant d 25. 1.
Bruhat intervals as rooks on skew Ferrers boards. Preprint, available at arXiv:math.CO/0601615
"... Abstract. We characterise the permutations π such that the elements in the closed lower Bruhat interval [id, π] of the symmetric group correspond to nontaking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id, π] corresponds to a flag m ..."
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Cited by 14 (0 self)
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Abstract. We characterise the permutations π such that the elements in the closed lower Bruhat interval [id, π] of the symmetric group correspond to nontaking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id, π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincaré polynomials (rankgenerating function) of Bruhat intervals with qrook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve qStirling numbers of the second kind. As a byproduct of our method, we present a new Stirling number identity connected to both Bruhat intervals and the polyBernoulli numbers defined by Kaneko. 1.
A 1Homotopy of Chevalley Groups
, 2008
"... In this paper, we describe the sheaves of A1homotopy groups of a simplyconnected Chevalley group G. The A1homotopy group sheaves can be identified with the sheafification of the unstable KaroubiVillamayor Kgroups. ..."
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Cited by 10 (4 self)
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In this paper, we describe the sheaves of A1homotopy groups of a simplyconnected Chevalley group G. The A1homotopy group sheaves can be identified with the sheafification of the unstable KaroubiVillamayor Kgroups.