a characterization of the finite simple groups abstract in this paper we obtain a quantitative characterization of all finite simple groups let t g denote the set of indices of maximal subgroups of group g and let p g be the smallest number in t g we have the following theorems theorem 2 let n and g be finite simple groups if n divides g p n p g and t n t g then or and 2 11 or 2 and 3 theorem 3 let n be a finite simple and let g be a finite group if g n and t g t n then g n

low dimensional real lie algebras all real lie algebras of dimension up to 8 that admit a nontrivial levi decomposition are found

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

on the probability of generating a monolithic group abstract a group l is primitive monolithic if l has a unique minimal normal subgroup n and trivial frattini subgroup by p l n k we denote the conditional probability that k randomly chosen elements of l generate l given that they project onto generators for l n in this article we show that p l n k is controlled by p y s 2 where n s r and y is a 2 generated almost simple group with socle s that is contained in the normalizer in l of the first direct factor of n information about p l n k for l primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups

the evolution partial differential equation ut uxxx 3 uxxu2 3u2xu 3uxu4 the evolution equation ut uxxx 3 uxxu2 3u2xu 3uxu4 u u x t is integrable it can be exactly linearized by an appropriate change of dependent variable hence several explicit solutions of the partial differential equation pde can be exhibited some of them display a remarkable solitronic phenomenology

the factorizations of the finite exceptional groups of lie type a finite group is said to be factorizable if g ab for some proper subgroups a and b of g such an expression is called a factorization of g factorizations have been the subject of considerable study see for example 11 121 for the case where g is a finite simple group some results can be found in 8 20 26 27 291 in this paper we determine completely the factorizations of the exceptional simple groups of lie type and their automorphism groups here by an exceptional simple group of lie type we mean a non abelian finite simple group associated with one of the families g f4 e e e b 2g2 3d f4 and 2e excluding g 2 and g 3 in view of the isomorphisms g 2 e u 3 and g 3 z l 8 there are two interesting features of our main result one is that remarkably few of the exceptional simple groups of lie type are factorizable for example e q ec q e q and e q possess no factorizations in contrast some striking factorizations are found to exist for the groups g q and f q for g q these factorizations are already known 20 271 in another paper 151 the maximal factorizations of all the other finite simple groups and their automorphism groups are classified where the factorization g ab is maximal if both a and b are maximal subgroups of g the results of this paper and 151 are used in 161 to obtain a classification of the maximal subgroups of the finite alternating and symmetric groups our main results here are the following two theorems

generation of finite classical groups by pairs of elements with large fixed point spaces abstract we study good elements in finite 2 n dimensional classical groups g namely t is a good element if o t is divisible by a primitive prime divisor of q n 1 for the relevant field order q and t fixes pointwise an n space the group sl 2 n q contains such elements and they are present in su 2 n q sp 2 n q so 2 n e q only if n is odd even even respectively we prove that there is an absolute positive constant c such that two random conjugates of t generate g with probability at least c if g so 2 n e 2 and g sp 2 n q with q even in the exceptional case g sp 2 n q with q even two conjugates of t never generate g in this case we prove that two random conjugates of t generate a subgroup so 2 n e q with probability at least c the results underpin analysis of new constructive recognition algorithms for classical groups in even characteristic which succeed where methods utilising involution centralisers are not available

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

comment on group analysis of kdv equation with time dependent coefficients abstract we show that the group classification results of the article entitled group analysis of kdv equation with time dependent coefficients which appeared in a g johnpillai m c khalique group analysis of kdv equation with time dependent coefficients appl math comput 216 2010 3761 3771 can be obtained from those of a more general class by a change of variables

finitary lie algebras abstract an algebra is called finitary if it consists of finite rank transformations of a vector space we classify finitary simple and finitary irreducible lie algebras over an algebraically closed field of characteristic 2 3

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

lie symmetries versus integrability in evolution equations motivated by some recent results obtained concerning the invariance properties of a particular family of generalized kdv equations we investigate the possible significance of classical lie invariance groups as a test for complete integrability

2 weights for unitary groups this paper gives a description of the local structures of 2 radical subgroups in a finite unitary group and proves alperin s weight conjecture for finite unitary groups when the characteristic of modular representation is even

on thompsonʼs conjecture for some finite simple groups in this article we prove a conjecture of j g thompson for an infinite class of simple groups of lie type more precisely we show that every finite group g with the property z g 1 and cs g cs psln q is necessarily isomorphic to psln q note that cs g is the set of lengths of conjugacy classes of g

distance transitive graphs and finite simple groups introductionthis paper represents the first step in the classification of finite primitive distancetransitive graphs in it we reduce the problem to the case where the automorph ism group is either almost simple or affine let be a simple connected undirected graph with verte oc x se t q e q i fthen d a j8 denotes the distance between 3 a i ann

linearizability for third order evolution equations the problem of linearization for third order evolution equations is considered criteria for testing equations for linearity are presented a class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite dimensional symmetry group linearizing transformations for this class are found using symmetry structure and local conservation laws a number of special cases as examples are discussed their transformation to equations within the same class by differential substitutions and connection with kdv and mkdv equations is also reviewed in this framework

infinite dimensional symmetries of two dimensional generalized burgers equations the conditions for a class of generalized burgers equations which a priori involve nine arbitrary functions of one or two variables to allow an infinite dimensional symmetry algebra are determined although this algebra can involve up to two arbitrary functions of time it does not allow a virasoro subalgebra this result reconfirms a long standing fact that variable coefficient generalizations of a nonintegrable equation should be expected to remain as such

new scale invariant nonlinear differential equations for a complex scalar field abstract we describe all complex wave equations of the form u f u u invariant under the extended poincare group as a result we have obtained the five new classes of p 1 3 invariant nonlinear partial differential equations for the complex scalar field

fuchsian groups finite simple groups and representation varieties let be a fuchsian group of genus at least 2 at least 3 if is non oriented we study the spaces of homomorphisms from to finite simple groups g and derive a number of applications concerning random generation and representation varieties precise asymptotic estimates for hom g are given implying in particular that as the rank of g tends to infinity this is of the form g 1 o 1 where is the measure of we then prove that a randomly chosen homomorphism from to g is surjective with probability tending to 1 as g combining our results with lang weil estimates from algebraic geometry we obtain the dimensions of the representation varieties text hom gamma bar g where bar g is gl n k or a simple algebraic group over k an algebraically closed field of arbitrary characteristic a key ingredient of our approach is character theory involving the study of the zeta function g s 1 s where the sum is over all irreducible complex characters of g

quadratic approximation of solutions for differential equations with nonlinear boundary conditions the quasilinearization method is used for nonlinear ordinary differential equations with nonlinear boundary conditions given are sufficient conditions when corresponding monotone sequences converge to the unique solution and this convergence is quadratic

symmetries of discrete dynamical systems involving two species the lie point symmetries of a coupled system of two nonlinear differential difference equations are investigated it is shown that in special cases the symmetry group can be infinite dimensional in other cases up to ten dimensional the equations can describe the interaction of two long molecular chains each involving one type of atoms

large subgroups of simple groups abstract let g be a finite group a proper subgroup h of g is said to be large if the order of h satisfies the bound h 3 g in this note we determine all the large maximal subgroups of finite simple groups and we establish an analogous result for simple algebraic groups in this context largeness is defined in terms of dimension an application to triple factorisations of simple groups both finite and algebraic is discussed

galilei symmetries of kdv type nonlinear evolution equations abstract we perform galilei symmetry group classification of a class of third order nonlinear evolution equations in one spatial variable which generalize kdv and mkdv equations all inequivalent pdes belonging to the class in question which admit the classical galilei group the extended galilei group and the natural extensions of the extended galilei groups are constructed the list of so obtained invariant equations may be used as motion equations for they all satisfy the galilei relativity principle in addition we also give a complete classification of group invariant solutions for one obtained galilei invariant equation

a family of wave equations with some remarkable properties we consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions recursion operators are found for two members of the family investigated for one of them a lax pair is also obtained proving its complete integrability from the lax pair we construct a miura type transformation relating the original equation to the korteweg de vries kdv equation this transformation on the other hand enables us to obtain solutions of the equation from the kernel of a schrodinger operator with potential parametrized by the solutions of the kdv equation in particular this allows us to exhibit a kink solution to the completely integrable equation from the 1 soliton solution of the kdv equation finally peakon type solutions are also found for a certain choice of the parameters although for this particular case the equation is reduced to a homogeneous second order nonlinear evolution equation

left symmetric structures on simple modular lie algebras abstract this paper investigates left symmetric structures on finite dimensional simple lie algebras g over a field k if k is of characteristic 0 then g does not admit any left symmetric structure this is known in the theory of affine manifolds in the modular case however such structures may exist the main purpose of this paper is to show that classical simple lie algebras of characteristic p 3 admit left symmetric structures only in case p divides dim g the proof involves the computation of the first cohomology groups of classical lie algebras for certain g modules of small dimension here g is regarded as the lie algebra of a connected semisimple algebraic group over an algebraically closed field of characteristic p 0 most of these computations are due to jantzen for nonrestricted simple lie algebras of cartan type it is shown that many more left symmetric structures can be found one studies so called adjoint structures induced by nonsingular derivations of g the simple algebra l g of block of dimension pn 1 for example admits adjoint structures for every p 0 if p 2 the results are more complicated

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

character ratios representation varieties and random generation of finite groups of lie type abstract we use character theory of finite groups of lie type to establish new results on representation varieties of fuchsian groups and also on probabilistic generation of groups of lie type

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

generalized kadomtsev petviashvili equation with an infinite dimensional symmetry algebra abstract a generalized kadomtsev petviashvili equation describing water waves in oceans of varying depth density and vorticity is discussed a priori it involves 9 arbitrary functions of one or two variables the conditions are determined under which the equation allows an infinite dimensional symmetry algebra this algebra can involve up to three arbitrary functions of time it depends on precisely three such functions if and only if it is completely integrable

simple groups probabilistic methods and a conjecture of kantor and lubotzky abstract we prove that a randomly chosen involution and a randomly chosen additional element of a finite simple groupggenerategwith probability 1 as g this confirms a conjecture of kantor and lubotzky applications and related results are derived for example we show that except for the suzuki groups and finitely many possible other exceptions all finite simple groups can be generated by two elements one of which has order 3 we also obtain sharp estimates on the probabilityp g of generating a finite simple group of exceptional lie typegby two randomly chosen elements this complements analogous estimates of babai and kantor for alternating and classical groups denoting bym g the minimal index of a proper subgroup of a finite simple groupg we conclude in particular thatm g 1 p g is bounded between two positive absolute constants

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

symmetries of variable coefficient korteweg de vries equations the lie point symmetries of the equation ut f x t uux g x t uxxx 0 are studied the symmetry group is shown to be at most four dimensional and this occurs if and only if the equation is equivalent under local point transformations to the kdv equation with f g 1 for nine different classes of functions f and g the symmetry group turns out to be three dimensional two dimensional and one dimensional symmetry groups occur for 11 and 15 classes of equations respectively

on the maximum orders of elements of finite almost simple groups and primitive permutation groups we determine upper bounds for the maximum order of an element of a finite almost simple group with socle t in terms of the minimum index m t of a maximal subgroup of t for t not an alternating group we prove that with finitely many excep tions the maximum element order is at most m t moreover apart from an explicit list of groups the bound can be reduced to m t 4 these results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n 4 we note again that this result gives upper bounds for meo aut t in terms of m t and for meo g in terms of m g since m t m g moreover equality in the up per bound meo aut t m t holds when t psld q for all but two pairs d q see table 3 and theorem 2 16 theorem 2 16 and table 3 provide good estimates for

preliminary group classification of the nonlinear differential difference equations abstract preliminary group classification of the nonlinear differential difference equations of the form u n f n t u n 1 u n u n 1 is considered by using the classical infinitesimal lie method and the theory of classification of abstract low dimensional lie algebras restrict to lie point symmetries and assume that the lie algebra of the symmetry group is realized by a particular vector field it is proved that there is only one equation admitting three dimensional simple lie algebras what is more all the inequivalent equations admitting semi simple lie algebras are nothing but the one also there exist inequivalent equations admitting low dimensional solvable lie algebras and semi direct sums of semi simple and solvable lie algebras

the maximal subgroups of the chevalley groups g2 q with q odd the ree groups 2g2 q and their automorphism groups

transitive subgroups of primitive permutation groups abstract we investigate the finite primitive permutation groups g which have a transitive subgroup containing no nontrivial subnormal subgroup of g the conclusion is that such primitive groups are rather rare and that their existence is intimately connected with factorisations of almost simple groups a corollary is obtained on primitive groups which contain a regular subgroup heavily involved in our proofs are some new results on subgroups of simple groups which have orders divisible by various primes for example another corollary implies that for every simple group t apart from l 3 3 u 3 3 and l 2 p with p a mersenne prime there is a collection consisting of two or three odd prime divisors of t such that if m is a subgroup of t of order divisible by every prime in then m is divisible by all the prime divisors of t and we obtain a classification of such subgroups m

equivalence groupoid of a class of variable coefficient korteweg de vries equations we classify the admissible transformations in a class of variable coefficient korteweg de vries equations as a result a full description of the structure of the equivalence groupoid of the class is given the class under study is partitioned into six disjoint normalized subclasses the widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes the group classification of one of the subclasses is carried out as an illustrative example

periodic simple groups of finitary linear transformations a group is locally finite if every finite subset generates a finite subgroup a group of linear transformations is finitary if each element minus the identity is an endomorphism of finite rank the classification and structure theory for locally finite simple groups splits naturally into two cases those groups that can be faithfully represented as groups of finitary linear transformations and those groups that are not finitary linear this paper completes the finitary case we classify up to isomorphism those infinite locally finite simple groups that are finitary linear but not linear

affine planes admitting a collineation group with a transitive action on the line at infinity abstract the affine planes admitting a collineation group with a transitive action on the line at infinity are investigated and an essentially complete classification is achieved when the group is almost simple a rather deep result is also obtained when the action on the line at infinity is faithful and primitive

classical groups probabilistic methods and the 2 3 generation problem we study the probability that randomly chosen elements of prescribed type in a finite simple classical group g generate g in particular we prove a conjecture of kantor and lubotzky in this area the probabilistic approach is then used to determine the finite simple classical quotients of the modular group psl2 z up to finitely many exceptions

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

ℵ0 categorical structures smoothly approximated by finite substructures a classification is given of primitive k0 categorical structures which are smoothly approximated by a chain of finite homogeneous substructures the proof uses the classification of finite simple groups and some representation theory the main theorems give information about a class of structures more general than the x0 categorical co stable structures examined by cherlin harrington and lachlan

equivalence groupoid and group classification of a class of variable coefficient burgers equations abstract we study admissible transformations and lie symmetries for a class of variable coefficient burgers equations we combine the advanced methods of splitting into normalized subclasses and of mappings between classes that are generated by families of point transformations parameterized by arbitrary elements of the original classes a nontrivial differential constraint on the arbitrary elements of the class of variable coefficient burgers equations leads to its partition into two subclasses which are related to normalized classes via families of point transformations parameterized by subclasses arbitrary elements one of the mapped classes is proved to be normalized in the extended generalized sense and its effective extended generalized equivalence group is found using the mappings between classes and the algebraic method of group classification we carry out the group classification of the initial class with respect to its equivalence groupoid

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

symmetry classification of diatomic molecular chains a symmetry classification of possible interactions in a diatomic molecular chain is provided for nonlinear interactions the group of lie point transformations leaving the lattice invariant and taking solutions into solutions is at most five dimensional an example is considered in which subgroups of the symmetry group are used to reduce the dynamical differential difference equations to purely difference ones

the 2 transitive ovoids an ovoid 0 in a classical polar space is a set of singular points such that every maximal totally singular subspace contains just one point in lo ovoids are intimately connected with other combinatorial objects including translation planes spreads partial geometries codes generalized hexagons and kerdock sets see s 9 163 for example interestingly enough many of the known ovoids are in fact 2 transitive that is they admit a 2 transitive automorphism group the notion of the automorphism group is made precise in 2 3 for example the suzuki groups sz q and the ree groups g q act 2 transitively on ovoids in 4 dimensional symplectic geometry and 7 dimensional orthogonal geometry respectively furthermore the unitary groups psu q for suitable prime powers q and the linear groups psl q3 with q even act 2 transitively on ovoids in 7or 8 dimensional orthogonal geometry the occurrence of such a large number of 2 transitive ovoids suggests that a classification of them is worthwhile much in the same spirit as kantor s classification of the finite linear spaces whose automorphism group acts 2 transitively on points lo the classification of the 2 transitive ovoids appears as our main theorem in section 2 we discover no new ovoids however we obtain some new results concerning the number of isomorphism classes of 2 transitive ovoids see section 2 our proof relies on the classification of the finite 2 transitive permutation groups which in turn relies on the recent classification of finite simple groups wc also draw upon several facts from the modular representation theory of finite groups

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 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

group analysis of general burgers korteweg de vries equations the complete group classification problem for the class of 1 1 dimensional rth order general variable coefficient burgers korteweg de vries equations is solved for arbitrary values of r greater than or equal to two we find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations showing that this class and certain gauged subclasses are normalized in the usual sense we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass and it is the latter problem that is solved efficiently using the algebraic method of group classification similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable respectively applying an original technique we classify lie reductions of equations from the class under consideration with respect to its equivalence group studying alternative gauges for eq

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

generation of finite simple groups by an involution and an element of prime order abstract we prove that every non abelian finite simple group is generated by an involution and an element of prime order

symmetry classification of kdv type nonlinear evolution equations group classification of a class of third order nonlinear evolution equations generalizing kdv and mkdv equations is performed it is shown that there are two equations admitting simple lie algebras of dimension three next we prove that there exist only four equations invariant with respect to lie algebras having nontrivial levi factors of dimension four and six our analysis shows that there are no equations invariant under algebras which are semi direct sums of levi factor and radical making use of these results we prove that there are three nine thirty eight fifty two inequivalent kdv type nonlinear evolution equations admitting one two three and four dimensional solvable lie algebras respectively finally we perform a complete group classification of the most general linear third order evolution equation