generation and random generation from simple groups to maximal subgroups abstract let g be a finite group and let d g be the minimal number of generators for g it is well known that d g 2 for all non abelian finite simple groups we prove that d h 4 for any maximal subgroup h of a finite simple group and that this bound is best possible we also investigate the random generation of maximal subgroups of simple and almost simple groups by applying a recent theorem of jaikin zapirain and pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant we then apply our results to the study of permutation groups in particular we show that if g is a finite primitive permutation group with point stabilizer h then d g 1 d h d g 4

fast recognition of alternating groups of unknown degree abstract we present a constructive recognition algorithm to decide whether a given black box group is isomorphic to an alternating or a symmetric group without prior knowledge of the degree this eliminates the major gap in known algorithms as they require the degree as additional input our methods are probabilistic and rely on results about proportions of elements with certain properties in alternating and symmetric groups these results are of independent interest for instance we establish a lower bound for the proportion of involutions with small support

involutive systems of differential equations einstein s strength versus cartan s degre d arbitraire three new theorems relating einstein s notions of strength and compatibility to the field of the initial value problem are presented these theorems result i in a first proof of matthews conjectures concerning this relation for a wider class of systems of partial differential equations ii in a new interpretation of einstein s compatibility condition and iii in the exact relation between einstein s strength and cartan s degre d arbitraire

large element orders and the characteristic of lie type simple groups we show that the three largest element orders determine the characteristic of lie type simple groups of odd characteristic this result was motivated by algorithmic needs in computations with matrix groups

fixed point ratios in actions of finite classical groups iii abstract this is the first in a series of four papers on fixed point ratios in actions of finite classical groups our main result states that if g is a finite almost simple classical group and is a faithful transitive non subspace g set then either fpr x x g 1 2 for all elements x g of prime order or g is one of a small number of known exceptions here fpr x denotes the proportion of points in which are fixed by x in this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups a further application concerning monodromy groups of covers of riemann surfaces is also outlined

transitive projective planes and insoluble groups suppose that a group g acts transitively on the points of p a finite non desarguesian projective plane we prove first that the sylow 2subgroups of g are cyclic or generalized quaternion we then prove that if g is insoluble then g o g is isomorphic to sl2 5 or sl2 5 2 msc 2000 20b25 51a35

overgroups in gl nr f of certain subgroups of sl n k i for any pair of division rings k and f with k f and dimfk r we determine all the overgroups of sl n k in gl nr f as well as the overgroups of sp n k in gl nr f for commutative k and even n the overgroups of su n k f and n k q in gl nr f will be determined in another paper overgroups in gl nr f of certain subgroups of sl n k ii

on the strength of maxwell s equations the strength of a set of field equations first defined by einstein as the number of taylor coefficients of field variables that could be chosen arbitrarily is used to show that the amount of initial data required by the electromagnetic formulation of maxwell s theory in free space is equal without approximation to that required by the potential formulation in each formulation the strength is interpreted in terms of the amount of initial data required to provide a solution of the cauchy initial value problem and in terms of the invariance properties of the formulation equality of the strengths of the two formulations of maxwell s theory is used to support the assertion that knowledge of the strengths of other established field theories provides a means for predicting the possible existence of unknown formulations of the theories

subquadratic time factoring of polynomials over finite fields new probabilistic algorithms are presented for factoring univariate polynomials over finite fields the algorithms factor a polynomial of degree n over a finite field of constant cardinality in time o n 1 815 previous algorithms required time n 2 o 1 the new algorithms rely on fast matrix multiplication techniques more generally to factor a polynomial of degree n over the finite field f q with q elements the algorithms use o n 1 815 log q arithmetic operations in f q the new baby step giant step techniques used in our algorithms also yield new fast practical algorithms at super quadratic asymptotic running time and subquadratic time methods for manipulating normal bases of finite fields

a note on the solvability of groups abstract let m be a maximal subgroup of a finite group g and k l be a chief factor such that l m while k m we call the group m k l a c section of m and we define sec m to be the abstract group that is isomorphic to a c section of m for every maximal subgroup m of g assume that sec m is supersolvable then any composition factor of g is isomorphic to l 2 p or z q where p and q are primes and p 1 mod 8 this result answer a question posed by y wang s li c sections of maximal subgroups of finite groups j algebra 229 2000 86 94

constructive recognition of classical groups in even characteristic abstract let g x gl d f be a classical group in its natural representation defined over a finite field f of even characteristic we present las vegas algorithms to construct standard generators for g which permit us to write an element of g as a straight line program in x and to construct an involution as a straight line program in x if f 4 then the algorithms run in time polynomial in the size of the input subject to the existence of a discrete logarithm oracle for f

cyclic quotients of transitive groups abstract let a be a transitive subgroup of s n we show that the largest cyclic quotient of a has order at most n this can be interpreted as an equivalent result about extensions of constants in the galois closure of a covering of curves over a finite field we also prove that the point stabilizer in a finite primitive permutation group always has a faithful orbit

exterior differential systems janet riquier theory and the riemann lanczos problems in two three and four dimensions we discuss the riemann lanczos problems in two three and four dimensions using the theory of exterior differential systems and janet riquier theory we show that the riemann lanczos problem in two dimensions is always a system in involution for each of the two possible signatures we give the general solution in both instances and show that the occurrence of characteristic coordinates need not affect the result in three dimensions the riemann lanczos problem is not in involution as an identity occurs this does not prevent the existence of singular solutions and we give an example for the reduced godel space time a prolongation of this problem whereby an integrability condition is added leads to a prolonged system in involution the riemann lanczos problem in four dimensions is not in involution and needs to be prolongated as bampi and caviglia suggested but singular solutions of it can be found and we give examples for the godel kasner and debever hubaut space times

finite groups admitting grassmannian 4 designs abstract let v be a finite dimensional euclidean space we classify finite subgroups of go v that admit grassmannian 4 designs we also classify zariski closed subgroups g of go v over which sym 2 v 1 g is irreducible

the maximality of monomial subgroups of linear groups over division rings abstract the monomial subgroups are proved to be maximal in gl n k or sl n k for arbitrary division rings k with some exceptions

the primitive permutation groups of degree less than 2500 abstract in this paper we use the o nan scott theorem and aschbacher s theorem to classify the primitive permutation groups of degree less than 2500

a black box group algorithm for recognizing finite symmetric and alternating groups i we present a las vegas algorithm which for a given black box group known to be isomorphic to a symmetric or alternating group produces an explicit isomorphism with the standard permutation representation of the group this algorithm has applications in computations with matrix groups and permutation groups in this paper we handle the case when the degree n of the standard permutation representation is part of the input in a sequel we shall treat the case when the value of n is not known in advance as an important ingredient in the theoretical basis for the algorithm we prove the following result about the orders of elements of s n the conditional probability that a random element s n is an n cycle given that n 1 is at least 1 10

short presentations for alternating and symmetric groups we construct two kinds of presentations for the alternating and symmetric groups of degree n the first are on two generators in which the number of relations is o logn and the presentation length is o log 2 n the second have a bounded number of generators and relations and length o logn

chern simons terms in metric affine space time bianchi identities as euler lagrange equations the metric affine gauge theory of gravity encompasses a space time with the following geometrical fields coframe metric g and an independent linear connection within this geometrical framework all four forms b dc are constructed which qualify as boundary terms for a gauge lagrangian that is they are gl 4 r scalars as well as exact forms derived from chern simons type three forms c the result of our search is summarized in eq 4 20 the translational piece dctt is new the boundary terms effectively serve as lagrangians for the bianchi identities of nonmetricity torsion and curvature in the canonical formalism the normal parts of the chern simons three forms represent generating functions that are capable of generating new ashtekar type variables eventually the bach lanczos identity is generalized to the metric affine space time

topological generation of exceptional algebraic groups abstract let g be a simple algebraic group over an algebraically closed field k and let c 1 c t be non central conjugacy classes in g in this paper we consider the problem of determining whether there exist g i c i such that g 1 g t is zariski dense in g first we establish a general result which shows that if is an irreducible subvariety of g t then the set of tuples in generating a dense subgroup of g is either empty or dense in in the special case c 1 c t by considering the dimensions of fixed point spaces we prove that this set is dense when g is an exceptional algebraic group and t 5 assuming k is not algebraic over a finite field in fact for g g 2 we only need t 4 and both of these bounds are best possible as an application we show that many faithful representations of exceptional algebraic groups are generically free we also establish new results on the topological generation of exceptional groups in the special case t 2 which have applications to random generation of finite exceptional groups of lie type in particular we prove a conjecture of liebeck and shalev on the random r s generation of exceptional groups

maximal subgroups in finite and profinite groups we prove that if a finitely generated profinite group g is not generated with positive probability by finitely many random elements then every finite group f is obtained as a quotient of an open subgroup of g the proof involves the study of maximal subgroups of profinite groups as well as techniques from finite permutation groups and finite chevalley groups confirming a conjecture from ann of math 137 1993 203 220 we then prove that a finite group g has at most igic maximal soluble subgroups and show that this result is rather useful in various enumeration problems

parallel multisplitting explicit aor methods for numerical solutions of semilinear elliptic boundary value problems abstract a class of parallel multisplitting explicit aor methods for a large scale system of nonlinear algebraic equations which is a finite difference approximation of a semilinear elliptic boundary value problem are presented this class of methods avoid the inner iteration and are shown to converge monotonically either from above or from below to a solution of the system without the monotonicity property of nonlinearity moreover this class of methods are applicable to the pure neumann boundary value problems a sufficient condition for the uniqueness of the solutions is provided the global convergence of the methods and the influence of the acceleration factor on the convergence rate are considered the applications and numerical results are given

two point stabilisers of partition actions of linear groups a partition of a kl dimensional vector space v is a set of l subspaces each of dimension k such that their direct sum is the original space v in this paper we show that unless l 2 the action of a group l such that psl v l pgl v on the set of partitions of v into l subspaces of dimension k is base two there exist two partitions v and w such that l v w 1 we will also show that given any finite group g there exist k l and partitions v w such that l v w g these results complement work the author has done with partition actions of the symmetric groups

on the convergence of the parallel multisplitting aor algorithm abstract we present a class of relaxed parallel multisplitting algorithms called the parallel multisplitting aor algorithm for solving large nonsingular systems of equations ax b this new algorithm is a generalization and improvement of the relaxed parallel multisplitting method linear algebra appl 119 141 152 1989 based on the new algorithm model we establish another algorithm called the relaxed parallel multisplitting aor algorithm the convergence of these algorithms is discussed under the condition that a is a monotone matrix we obtain corresponding convergence results these convergence conditions are convenient to verify

testing matrix groups for primitivity abstract we describe an algorithm which seeks to decide whether or not a matrix group defined over a finite field acts to preserve blocks of imprimitivity and if so to find a block system implementations of the algorithm are publicly available

black box exceptional groups of lie type if a black box group is known to be isomorphic to an exceptional simple group of lie type of twisted rank 1 other than any f4 q over a field of known size a las vegas algorithm is given to produce a constructive isomorphism in view of its timing this algorithm yields an upgrade of all known nearly linear time monte carlo permutation group algorithms to las vegas algorithms when the input group has no composition factor isomorphic to any group f4 q or g2 q

constructive recognition of finite alternating and symmetric groups acting as matrix groups on their natural permutation modules we present a las vegas algorithm which for a given matrix group known to be isomorphic modulo scalars to a finite alternating or symmetric group acting on the fully deleted permutation module produces an explicit isomorphism with the standard permutation representation of the group this algorithm exploits information available from the matrix representation and thereby is faster than existing black box recognition algorithms applied to these groups in particular it uses the fact that certain types of elements in these groups can be identified and constructed from the structure of their characteristic polynomials the algorithm forms part of a large scale program for computing with groups of matrices over finite fields when combined with existing black box recognition algorithms the results of this paper prove that any d dimensional absolutely irreducible matrix representation of a finite alternating or symmetric group over a finite field can be recognised with o d1 2 random group elements and o d1 2 matrix multiplications up to some logarithmic factors

a computer assisted analysis of some matrix groups abstract we use algorithms developed recently for the study of linear groups to investigate a sequence of matrix groups defined over gf 2 these are images of representations of certain finitely presented groups considered by soicher in a study of simplicial complexes related to the suzuki sequence graphs

on the arbitrariness of the general solution of an involutive partial differential equation the relationship between the strength of a differential equation as introduced by einstein its cartan characters and its hilbert polynomial is studied using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems the problem of computing the number of arbitrary functions in the general solution is treated finally the effect of gauge symmetries is considered

parallel nonlinear aor method and its convergence abstract we set up a class of parallel nonlinear aor method in the sense of matrix multi splitting for solving the large scale system of nonlinear equations ax x b with a l rn nonsingular b rn and rn rn being continuously diagonal the global as well as the monotone convergence of this method is proved