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22
Linear Differential Operators for Polynomial Equations
"... Given a squarefree polynomial P 2 k 0 [x; y], k 0 a number eld, we construct a linear dierential operator that allows one to calculate the genus of the complex curve dened by P = 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k 0 , and calculate i ..."
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Given a squarefree polynomial P 2 k 0 [x; y], k 0 a number eld, we construct a linear dierential operator that allows one to calculate the genus of the complex curve dened by P = 0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k 0 , and calculate information concerning the Galois group of P over k 0 (x) as well as over k 0 (x).
The computation of Galois groups over function fields
, 1992
"... Abstract. Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1,t2,...,tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degre ..."
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Abstract. Symmetric function theory provides a basis for computing Galois groups which is largely independent of the coefficient ring. An exact algorithm has been implemented over Q(t1,t2,...,tm) in Maple for degree up to 8. A table of polynomials realizing each transitive permutation group of degree 8 as a Galois group over the rationals is included.
Coherent Configurations, Association Schemes and Permutation Groups
 Groups, Combinatorics and Geometry
, 2003
"... Coherent configurations are combinatorial objects invented for the purpose of studying finite permutation groups; every permutation group which is not doubly transitive preserves a nontrivial coherent configuration. However, symmetric coherent configurations have a much longer history, having b ..."
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Coherent configurations are combinatorial objects invented for the purpose of studying finite permutation groups; every permutation group which is not doubly transitive preserves a nontrivial coherent configuration. However, symmetric coherent configurations have a much longer history, having been used in statistics under the name of association schemes. The relationship between permutation groups and association schemes is quite subtle; there are groups which preserve no nontrivial association scheme, and other groups for which there is not a unique minimal association scheme. This paper gives a brief outline of the theory of coherent configurations and association schemes, and reports on some recent work on the connection between association schemes and permutation groups. 1 Coherent configurations This section contains the definitions of coherent configurations and of various specialisations (including association schemes), and their connection with finite permutation...
ON THE SQUAREFREE SIEVE
, 2004
"... A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a squarefree polynomial. ..."
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A squarefree sieve is a result that gives an upper bound for how often a squarefree polynomial may adopt values that are not squarefree. More generally, we may wish to control the behavior of a function depending on the largest square factor of P(x1,..., xn), where P is a squarefree polynomial.
The Product Replacement Graph On Generating Triples Of Permutations
, 2000
"... . We prove that the product replacement graph on generating 3tuples of An is connected for n 11. We employ an efficient heuristic based on [P1] which works significantly faster than brute force. The heuristic works for any group. Our tests were confined to An due to the interest in Wiegold's Co ..."
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. We prove that the product replacement graph on generating 3tuples of An is connected for n 11. We employ an efficient heuristic based on [P1] which works significantly faster than brute force. The heuristic works for any group. Our tests were confined to An due to the interest in Wiegold's Conjecture, usually stated in terms of T systems (see [P2]). Our results confirm Wiegold's Conjecture in some special cases and are related to the recent conjecture of Diaconis and Graham [DG]. The work was motivated by the study of the product replacement algorithm (see [CLMNO,P2]). Introduction Let G be a finite group, and let N k (G) be the set of generating ktuples (g) = (g 1 ; : : : ; g k ), where hg 1 ; : : : ; g k i = G. Define moves on N k (G) as follows: R \Sigma i;j : (g 1 ; : : : ; g i ; : : : ; g k ) ! (g 1 ; : : : ; g i \Delta g \Sigma1 j ; : : : ; g k ) L \Sigma i;j : (g 1 ; : : : ; g i ; : : : ; g k ) ! (g 1 ; : : : ; g \Sigma1 j \Delta g i ; : : : ; g k ) ) ...
Conjugacy classes in finite permutation groups via homomorphic images
 Math. Comp., posted on May
, 1999
"... Abstract. The lifting of results from factor groups to the full group is a standard technique for solvable groups. This paper shows how to utilize this approach in the case of nonsolvable normal subgroups to compute the conjugacy classes of a finite group. 1. ..."
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Abstract. The lifting of results from factor groups to the full group is a standard technique for solvable groups. This paper shows how to utilize this approach in the case of nonsolvable normal subgroups to compute the conjugacy classes of a finite group. 1.
Techniques for the Computation of Galois Groups
 In Algorithmic algebra and number theory
, 1997
"... This note surveys recent developments in the problem of computing Galois groups. Galois theory stands at the cradle of modern algebra and interacts with many areas of mathematics. The problem of determining Galois groups therefore is of interest not only from the point of view of number theory (fo ..."
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This note surveys recent developments in the problem of computing Galois groups. Galois theory stands at the cradle of modern algebra and interacts with many areas of mathematics. The problem of determining Galois groups therefore is of interest not only from the point of view of number theory (for example see the article [39] in this volume), but leads to many questions in other areas of mathematics. An example is its application in computer algebra when simplifying radical expressions [32]. Not surprisingly, this task has been considered in works from number theory, group theory and algebraic geometry. In this note I shall give an overview of methods currently used. While the techniques used for the identification of Galois groups were known already in the last century [26], the involved calculations made it almost impractical to do computations beyond trivial examples. Thus the problem was only taken up again in the last 25 years with the advent of computers. In this note we will restrict ourselves to the case of the base field Q. Most methods generalize to other fields like Q(t), Q p , IF p (t) or number fields. The results presented here are the work of many mathematicians. I tried to give credit by references wherever possible. 1
New constructions of groups without semiregular subgroups
 Comm. Algebra
, 2007
"... An elusive permutation group is a transitive permutation group with no fixed point free elements of prime order, or equivalently, no nontrivial semiregular subgroups. We provide several new constructions of elusive groups, some of which enable us to build elusive groups with new degrees. 1 ..."
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An elusive permutation group is a transitive permutation group with no fixed point free elements of prime order, or equivalently, no nontrivial semiregular subgroups. We provide several new constructions of elusive groups, some of which enable us to build elusive groups with new degrees. 1
GAP Computations Concerning Probabilistic Generation of Finite Simple Groups
, 2007
"... This is a collection of examples showing how the GAP system [GAP07] can be used to compute information about the probabilistic generation of finite almost simple groups. It includes all examples that were needed for the computational results in [BGK]. The purpose of this writeup is twofold. On the o ..."
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This is a collection of examples showing how the GAP system [GAP07] can be used to compute information about the probabilistic generation of finite almost simple groups. It includes all examples that were needed for the computational results in [BGK]. The purpose of this writeup is twofold. On the one hand, the computations are documented this way. On the other hand, the GAP code shown for the examples can be used as test input for automatic checking of the data and the functions used. A first version of this document had been accessible in the web in April 2006. The main difference to the current version is that the format of the GAP output was adjusted to the changed behaviour