representations of the symmetric group are reducible over simply transitive subgroups if the characteristic of f is zero this problem has been solved by saxl 16 an important feature of saxl s result is that a groupg as in the problem above is either 2 transitive or fixes a point i e is contained in some n 1 the caseg n 1 can then be settled using the branching rule and induction on the other hand an explicit list of k transitive groups for k 2 is available which can be used to complete the proof in characteristic zero see 16 for more details from now on we assume that p 0 in this case the problem is important for determining maximal subgroups of finite classical groups 1 13 however the situation is nowmore complicated for example to determine the pairs g d as above withg an one needs the mullineux conjecture 7 2 ifg is intransitive then up to a conjugation it is contained in a standard young subgroup of the form n k k the irreducible restrictions from n to n 1 have been described in 14 see also 11 6 in 11 it is also

moving least squares particle hydrodynamics i consistency and stability the smooth particle hydrodynamics sph method is derived in a novel manner by means of a galerkin approximation applied to the lagrangian equations of continuum mechanics as in the finite element method this derivation is modified to replace the sph interpolant with the moving least squares mls interpolant of lancaster and saulkaskas and define a new particle volume which ensures thermodynamic compatibility a variable rank modification of the mls interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required a surprise benefit of the galerkin sph derivation is a theoretical justification of a common ad hoc technique for variable h sph the new mlsph method is conservative if an anti symmetric quadrature rule for the stiffness matrix elements can be supplied in this paper a simple one point collocation rule is used to retain similarity with sph leading to a non conservative method several examples document how mlsph renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the sph method spurious boundary effects erroneous rates of strain and rotation and tension instability two of these examples are non linear lagrangian patch tests with analytic solutions with which mlsph agrees almost exactly the examples also show that mlsph is not absolutely stable if the problems are run to very long times a linear stability analysis explains both why it is more stable than sph and not yet absolutely stable and an argument is made that for realistic dynamic problems mlsph is stable enough the notion of coherent particles for which the numerical stability is identical to the physical stability is introduced the new method is easily retrofitted into a generic sph code and some observations on performance are made copyright 1999 john wiley sons ltd

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

finite index subgroups of mapping class groups let g 3 and n 0 and let mg n be the mapping class group of a surface of genus g with n boundary components we prove that mg n contains a unique subgroup of index 2g 1 2g 1 up to conjugation a unique subgroup of index 2g 1 2g 1 up to conjugation and the other proper subgroups ofmg n are of index greater than 2g 1 2g 1 in particular the minimum index for a proper subgroup of mg n is 2g 1 2g 1 ams subject classification primary 57m99 secondary 20g40 20e28 0 introduction and statement of results the interaction between mapping class groups and finite groups has long been a topic of interest the famous hurwitz bound of 1893 showed that the mapping class group of a closed riemann surface of genus g has an upper bound of 84 g 1 for the order of its finite subgroups and kerckhoff showed that the order of finite cyclic subgroups is bounded above by 4g 2 19 20 the subject of finite index subgroups of mapping class groups was brought into focus by grossman s discovery that the mapping class group mg n 0 homeo g n of an oriented surface g n of genus g and n boundary components is residually finite and thus well endowed with subgroups of finite index 15 homeo g n denotes the space of those homeomorphisms of g n that preserve the orientation and are the identity on the boundary this prompts the dual question what is the minimum index mi mg n of a proper subgroup of finite index in mg n results to date have suggested that like the maximum finite order question the minimum index question should have an answer that is linear in g the best previously published bound is mi mg n 4g 4 for g 3 see 26 this inequality is used by aramayona and souto to prove that if g 6 and g 2g 1 then any nontrivial homomorphism mg n mg n is induced by an embedding 1 it is also an important ingredient in the proof of zimmermann 33 that for g 3 and 4 the minimal nontrivial quotient of mg 0 is sp2g f2 the headline result of this paper is the following exact exponential bound theorem 0 1 for g 3 and n 0 mi mg n mi sp2g z mi sp2g f2 2g 1 2g 1 this exponential bound is all the more surprising since in similar questions we get linear expected bounds for instance bridson 5 6 has proved that a mapping class group of a surface

disjoint pattern database heuristics we describe a new technique for designing more accurate admissible heuristic evaluation functions based on pattern databases j culberson j schaeffer comput intelligence 14 3 1998 318 334 while many heuristics such as manhattan distance compute the cost of solving individual subgoals independently pattern databases consider the cost of solving multiple subgoals simultaneously existing work on pattern databases allows combining values from different pattern databases by taking their maximum if the subgoals can be divided into disjoint subsets so that each operator only affects subgoals in one subset then we can add the pattern database values for each subset resulting in a more accurate admissible heuristic function we used this technique to improve performance on the fifteen puzzle by a factor of over 2000 and to find optimal solutions to 50 random instances of the twenty four puzzle

simulation of surface tension in 2d and 3d with smoothed particle hydrodynamics method the methods for simulating surface tension with smoothed particle hydrodynamics sph method in two dimensions and three dimensions are developed in 2d surface tension model the sph particle on the boundary in 2d is detected dynamically according to the algorithm developed by dilts g a dilts moving least squares particle hydrodynamics ii conservation and boundaries international journal for numerical methods in engineering 48 2000 1503 1524 the boundary curve in 2d is reconstructed locally with lagrangian interpolation polynomial in 3d surface tension model the sph particle on the boundary in 3d is detected dynamically according to the algorithm developed by haque and dilts a haque g a dilts three dimensional boundary detection for particle methods journal of computational physics 226 2007 1710 1730 the boundary surface in 3d is reconstructed locally with moving least squares mls method by transforming the coordinate system it is guaranteed that the interface function is one valued in the local coordinate system the normal vector and curvature of the boundary surface are calculated according to the reconstructed boundary surface and then surface tension force can be calculated surface tension force acts only on the boundary particle density correction is applied to the boundary particle in order to remove the boundary inconsistency the surface tension models in 2d and 3d have been applied to benchmark tests for surface tension the ability of the current method applying to the simulation of surface tension in 2d and 3d is proved

reduced fusion systems over p groups with abelian subgroup of index p ii abstract for an odd prime p we look at simple fusion systems over a finite nonabelian p group s which has an abelian subgroup a of index p when s has more than one such subgroup we reduce this to a case already studied by ruiz and viruel when a is the unique abelian subgroup of index p in s and is not essential equivalently is not radical in the fusion system we give a complete list of all possibilities which can occur this includes several families of exotic fusion systems including some which have proper strongly closed subgroups

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

irreducible restrictions of representations of symmetric and alternating groups in small characteristics abstract building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups such a classification is known when the characteristic of the ground field is greater than 3 but the small characteristics cases require a substantially more delicate analysis and new ideas our results fit into the aschbacher scott program on maximal subgroups of finite classical groups

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

some problems of wielandt revisited abstract several problems in the theory of finite permutation groups considered before by h wielandt are attacked by new and traditional methods one new method is given by the theorem that a semisimple subgroup a of a group g normalizing a different subgroup b isomorphic to a forces that the centralizer in ab of b is non trivial hence b is not the generalized fitting subgroup of its normalizer this theorem is applied in proving that the paired subconstituent g of a primitive permutation group g is faithful if the non trivial subconstituent g is regular if g is a non abelian simple group all of whose proper subgroups are solvable then the regularity of g even implies that g is faithful also several theorems are obtained for the case that a non trivial subconstituent g is nilpotent or more generally that g is subnormal in g for

breadth first heuristic search recent work shows that the memory requirements of a and related graph search algorithms can be reduced substantially by only storing nodes that are on or near the search frontier using special techniques to prevent node regeneration and recovering the solution path by a divide and conquer technique when this approach is used to solve graph search problems with unit edge costs we show that a breadth first search strategy can be more memory efficient than a best first strategy we also show that a breadth first strategy allows a technique for preventing node regeneration that is easier to implement and can be applied more widely the breadth first heuristic search algorithms introduced in this paper include a memory efficient implementation of breadth first branch and bound search and a breadth first iterative deepening a algorithm that is based on it computational results show that they outperform other systematic search algorithms in solving a range of challenging graph search problems

a characterization of solvability for finite groups in terms of their frame abstract the frame of a group is the poset of conjugacy classes of all its proper subgroups in this paper we will prove that a finite group is solvable if and only if every collection of maximal elements of its frame has a well defined meet and the poset consisting of all such meets including the meet of the empty set is a modular lattice

explicit surjectivity of galois representations for abelian surfaces and gl 2 varieties abstract let a be an absolutely simple abelian variety without potential complex multiplication defined over the number field k suppose that either dim a 2 or a is of gl 2 type we give an explicit bound l 0 a k such that for every prime l l 0 a k the image of gal k k in aut t l a is as large as it is allowed to be by endomorphisms and polarizations

linear time disk based implicit graph search many search algorithms are limited by the amount of memory available magnetic disk storage is over two orders of magnitude cheaper than semiconductor memory and individual disks can hold up to a terabyte we augment memory with magnetic disks to perform brute force and heuristic searches that are orders of magnitude larger than any previous such searches the main difficulty is detecting duplicate nodes which is normally done with a hash table due to long disk latencies however randomly accessed hash tables are infeasible on disk and are replaced by a mechanism we call delayed duplicate detection in contrast to previous work we perform delayed duplicate detection without sorting which runs in time linear in the number of nodes in practice using this technique we performed the first complete breadth first searches of the 2 7 3 5 4 times 4 and 2 8 sliding tile puzzles verifying the radius of the 4 4 puzzle and determining the radius of the others we also performed the first complete breadth first searches of the four peg towers of hanoi problem with up to 22 discs discovering a surprising anomaly regarding the radii of these problems in addition we performed the first heuristic searches of the four peg towers of hanoi problem with up to 31 discs verifying a conjectured optimal solution length to these problems we also performed partial breadth first searches of rubik s cube to depth ten in the face turn metric and depth eleven in the quarter turn metric confirming previous results

recognising the suzuki groups in their natural representations abstract under the assumption of a certain conjecture for which there exists strong experimental evidence we produce an efficient algorithm for constructive membership testing in the suzuki groups sz q where q 2 2 m 1 for some m 0 in their natural representations of degree 4 it is a las vegas algorithm with running time o log q field operations and a preprocessing step with running time o log q log log q field operations the latter step needs an oracle for the discrete logarithm problem in f q we also produce a recognition algorithm for sz q x this is a las vegas algorithm with running time o x 2 field operations finally we give a las vegas algorithm that given x h sz q for some h gl 4 q finds some g such that x g sz q the running time is o log q log log q x field operations implementations of the algorithms are available for the computer system magma

overgroups in gl u w of certain subgroups of gl u gl w i abstract let u w be two spaces over an arbitrary field f we have determined all the overgroups of sl u sl w in gl u w and the overgroups of sp u sp w in gl u w

a proof of the mullineux conjecture a partition of a positive integer n is a sequence 1 2 m 0 of integers such that i n for a positive integer p a partition 1 2 m or its young diagram is called p regular if it does not have p or more equal parts i e if there does not exist t m p 1 with t t 1 t p 1 let f be a field of characteristic p 0 it is well known that irreducible representations of the symmetric group sn over f are naturally parametrized by pregular partitions of n cf for example 9 12 if is such a partition we denote the corresponding irreducible module by d let sgnn be the one dimensional sign representation of sn over f i e sgnn f as a vector space and g f sgn g f for any g sn f f here sgn g is just the sign of the permutation g it is clear that for any irreducible d the tensor product d sgnn is also irreducible the problem usually called the problem of mullineux is to find the p regular partition such that d sgnn d put d sgnn dn in this way a bijection bn on the set pn of p regular partitions of n is defined for each positive integer n and the problem is

three dimensional boundary detection for particle methods the three dimensional exposure method for the detection of the boundary of a set of overlapping spheres is presented like the two dimensional version described in a previous paper the three dimensional algorithm precisely detects void opening or closure and is optimally suited to the kernel mediated interactions of smoothed particle hydrodynamics although it may be used in any application involving sets of overlapping spheres the principle idea is to apply the two dimensional method on the surface of each candidate boundary sphere to the circles of intersection with neighboring spheres as the algorithm finds the exact solution the quality of detection is independent of particle configuration in contrast to gradient based techniques the observed cpu execution times scale as o mn where m is the number of particles n is the average number of neighbors of a particle and is a problem dependent constant between 1 6 and 1 7 the time required per particle is comparable to the amount of time required to evaluate a three dimensional linear moving least squares interpolant at a single point

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

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

representations of the symmetric group which are irreducible over subgroups let f be an algebraically closed field of characteristic p and n be the symmetric group on n letters in this paper we classify all pairs g d where d is an irreducible f n module of dimension greater than 1 and g is a proper subgroup of n such that the restriction d g is irreducible provided p 3

irreducible restrictions of representations of symmetric groups in small characteristics reduction theorems we study irreducible restrictions of modules over symmetric groups to subgroups we get reduction results which substantially restrict the classes of subgroups and modules for which this is possible such results are known when the characteristic of the ground field is greater than 3 but the small characteristics cases require a substantially more delicate analysis and new ideas this work fits into the aschbacher scott program on maximal subgroups of finite classical groups

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

nonlinear dimensionality reduction by topologically constrained isometric embedding many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space the embedding process usually relies on distances between neighboring features mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to its non convexity distortions resulting from using long geodesics indiscriminately lead to a known limitation of the isomap algorithm when used to map non convex manifolds presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of flat manifolds with boundaries the resulting algorithm filters out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the boundary of the feature manifold thus avoiding an inherent limitation of the isomap algorithm since the proposed algorithm matches non local structures it is robust to strong noise we show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques both global and local in nature

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

arithmetic results on orbits of linear groups the first four authors acknowledge the support of an arc discovery grant the first author acknowledges the support of an australian research fellowship the third author acknowledges the support of a federation fellowship the fifth author acknowledges the support of the nsf grants dms 0901241 and dms 1201374

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

representations of the general linear groups which are irreducible over subgroups we classify all triples g v h such that sl n q leq g leq gl n q v is a representation of g of dimension greater than one over an algebraically closed field bbb f of characteristic prime to q and h is a proper subgroup of g such that the restriction v downarrow h is irreducible this problem is a natural part of the aschbacher scott program on maximal subgroups in finite classical groups

maximal overgroups of singer elements in classical groups abstract in this paper we use the results of a recent paper by guralnick penttila praeger and saxl proc london math soc 78 1999 167 214 to calculate an explicit list of the maximal overgroups of the singer elements in the finite special linear general unitary special unitary symplectic and minus type orthogonal groups

a generalized cameron kantor theorem abstract in this paper we prove that the non trivial unipotent characters of psl n q are reducible over any proper subgroup with the only exception of the unipotent character of psl 4 2 of degree 14 this result can be viewed as a generalization of a theorem of cameron and kantor

bounds on the number and sizes of conjugacy classes in finite chevalley groups with applications to derangements 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 boston shalev 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

primitive permutation groups with a solvable 2 transitive subconstituent for a permutation group g acting on a finite set and a point a suborbit is an orbit of the point stabilizer g on the permutation group induced by g on is called a subconstituent of g moreover g is said to be uniprimitive if g is primitive but not 2 transitive in this paper we investigate uniprimitive permutation groups which have a solvable 2 transitive subconstituent we determine all such groups g which have a simple socle the affine case that is g has an elementary abelian socle are also discussed and an infinite family of affine primitive groups with non self paired 2 transitive subconstituents are presented

primitive permutation groups of bounded orbital diameter 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

on fixed points of elements in primitive permutation groups abstract the fixity of a finite permutation group is the maximal number of fixed points of a non identity element we study the fixity of primitive groups of degree n showing that apart from a short list of exceptions the fixity of such groups is at least n 1 6 we also prove that there is usually an involution fixing at least n 1 6 points

factorizations of almost simple groups with a factor having many nonsolvable composition factors abstract this paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors this together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor

fast free surface detection and level set function definition in sph solvers the present paper proposes a novel algorithm to detect the free surface in particle simulations both in two and three dimensions since the proposed algorithms are based on sph interpolations their implementation does not require complex geometrical procedures thus the free surface detection can be easily embedded in sph solvers without a significant increase of the cpu time throughout this procedure accurate normal vectors to the free surface are made available then it is possible to define a level set function algorithm which is presented in detail the latter allows in depth analyses of three dimensional free surface simulations by using standard visualization tools including internal features of the flow the algorithms proposed for detecting free surface particles and defining the level set function are validated on simple and complex two and three dimensional flow simulations the usefulness of the proposed procedures to post process and analyze complex flows are illustrated on realistic examples

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

frontier search the critical resource that limits the application of best first search is memory we present a new class of best first search algorithms that reduce the space complexity the key idea is to store only the open list of generated nodes but not the closed list of expanded nodes the solution path can be recovered by a divide and conquer technique either as a bidirectional or unidirectional search for many problems frontier search dramatically reduces the memory required by best first search we apply frontier search to breadth first search of sliding tile puzzles and the 4 peg towers of hanoi problem dijkstra s algorithm on a grid with random edge costs and the a algorithm on the fifteen puzzle the four peg towers of hanoi problem and optimal sequence alignment in computational biology

on restrictions of modular spin representations of symmetric and alternating groups let f be an algebraically closed field of characteristic p and h be an almost simple group or a central extension of an almost simple group an important problem in representation theory is to classify the subgroups g of h and fh modules v such that the restriction v g is irreducible for example this problem is a natural part of the program of describing maximal subgroups in finite classical groups in this paper we investigate the case of the problem where h is the schur s double cover n or s n

representations of the alternating group which are irreducible over subgroups ii nsf dms 1161094 dms 0901241 dms 1201374 humboldt foundation simons foundation 204181 simons foundation fellowship 305247

depth first iterative deepening an optimal admissible tree search abstract the complexities of various search algorithms are considered in terms of time space and cost of solution path it is known that breadth first search requires too much space and depth first search can use too much time and doesn t always find a cheapest path a depth first iterative deepening algorithm is shown to be asymptotically optimal along all three dimensions for exponential tree searches the algorithm has been used successfully in chess programs has been effectively combined with bi directional search and has been applied to best first heuristic search as well this heuristic depth first iterative deepening algorithm is the only known algorithm that is capable of finding optimal solutions to randomly generated instances of the fifteen puzzle within practical resource limits

overgroups of primitive groups ii we continue our study of the overgroup lattices of subgroups of finite alternating and symmetric groups with applications to the question of palfy and pudlak as to whether each finite lattice is an interval in the lattice of subgroups of some finite group

computing matrix group decompositions with respect to a normal subgroup abstract we describe an algorithm which can be used to investigate whether a matrix group defined over a finite field decomposes with respect to a normal subgroup defined as the normal closure of a given set of matrices the possible decompositions correspond to classes in aschbacher s classification of subgroups of the general linear group

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 restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups i 1994 on restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups ii communications in algebra vol 22 no 15 pp 6175 6208

a recognition algorithm for special linear groups contain th speciase l linea sl rfr grou q pamplifying the problem he said he would be happy to have a program that couldbe used fo r genera l purpose practical computatio in th range e rf n 30 charle ssims suggested tha onte should be more ambitiou ass to degree sa y d 70 bu trestrict onesel tfo the case q 2 in this pape wr e shall propos ae method whichwe believ teo be practica fol r value osf d up t somewhero e aroun 60d or 70 andwhich is no particularlt y sensitiv to thee size of q w muse t emphasize howeverthat our ideas hav noe t yet at th time e of writing been implemente andd tested until adequate evidenc ha accumulates e d such claim nost mus b takee t n veryseriously ou algorithr ims partly probabilistic it whes respons n ies positiv iettells us that the group generate bdy x certainly does contai sl rfn q but if theoutput is negative that assertion will usually b onle mady e with probabilit 1y e where e is a pre determined small positive number the analogous proble fomr permutatio n group is tso decide fo r a given se t xof permutation osf a set o sizf e n whethe thr e subgroup generate bdy x containsthe alternating group alt n an elegan t probabilistic algorith fo recognizinr m galternating and symmetric group hass been in use fo manr y years le t u cals l anelement of th symmetrie c group sym n if on primae of cycle its has prims elength p non oef the othe r cycles have length divisibl by p ane d ps n 3 themethod exploit thse following facts i if g i alt zs or sym n the an good proportio onf th elemente s o f g areprima ii if g does no contait n alt n and if g i primitivs a e permutation groupthen g contains no such elements the forme irs no hart d to prove see fo exampler 20 lectur e 11 th lattee rassertion comes from a well known old theorem of jordan see fo exampler 30 theorem 13 9 accordingl the ide ya i s to tes it f g i primitives whic cahn bedone pretty quickl byy fo exampler a method of atkinson i f no thet n certainlyg is neither alt n no sym n r i f g i primitives th e crucial ste isp to make asuitable numbe orf independent random selection of elements osf g no mant yare needed see fo r exampl 20 e som if e prima permutation show up thens gis identified correctl as alt ny o r sym n i f not the n with high probabilit g ydoes not contain alt n thus ther is a smale l probability tha the algoritht m willfail to identify alt n or sym n this procedur wa firsts e programme ford

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

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

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

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