prime characters and primitivity a natural question in the character theory of finite groups is which quasi primitive characters are actually primitive a character is said to be quasi primitive if it is irreducible and its restriction to every normal subgroup is homogeneous if the group is solvable t berger 9 th 11 331 has shown that every quasi primitive character is actually primitive if the group is simple however every irreducible character is quasi primitive and may or may not be primitive the character of degree 5 of alt 5 for example is not primitive it follows that a generalization of berger s theorem to non solvable groups must include some extra hypotheses which become vacuous when the group is solvable in this paper we prove the following such theorem with hypotheses on the chief factors and the character degrees

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

the geometric foundations of the integrability property of differential equations and physical systems ii mechanics on affinely connected manifolds and the work of kowalewski and painleve in the 1880 s sophie kowalewski proposed to study integrability of differential equations in terms of the analyticity properties of their solutions the model she studied the spinning top moving under gravity has an interesting geometric and algebraic structure which will be generalized the differential geometric properties of such models will be investigated in terms of the theory of affinely connected manifolds and lie theory

an integrable sis model we provide a demonstration of the integrability of a classical model of an infectious disease which neither kills nor induces autoimmunity by means of the painleve analysis and use the lie theory of transformation groups to present an explicit solution

simple classical groups of lie type are determined by their character degrees let g be a finite group denote by irr g the set of all irreducible complex characters of g let cd g be the set of all irreducible complex character degrees of g forgetting multiplicities that is cd g 1 irr g and let x1 g be the set of all irreducible complex character degrees of g counting multiplicities let h be a finite nonabelian simple classical group in this paper we will show that if g is a finite group and x1 g x1 h then g is isomorphic to h in particular this implies that the nonabelian simple classical groups of lie type are uniquely determined by the structure of their complex group algebras

on the automorphism groups of quasiprimitive almost simple graphs let gamma be a graph and let g be a subgroup of automorphisms of gamma then g is said to be locally primitive on gamma if for each vertex upsilon the stabilizer g upsilon induces a primitive group of permutations on the set of vertices adjacent to upsilon this paper investigates pairs gamma g for which g is locally primitive on gamma g is an almost simple group that is l less than or equal to g less than or equal to aut l for some nonabelian simple group l and the simple socle l is transitive on vertices each such graph is a cover of a possibly smaller graph l gamma over tildeg on which g is also locally primitive and for which in addition aut l gamma over tildeg is quasiprimitive on vertices that is every nontrivial normal subgroup of aut l gamma over tildeg is vertex transitive it is proved that aut l gamma over tildeg is also an almost simple group tn the general case in which aut gamma is not quasiprimitive on vertices we show that either every intransitive minimal normal subgroup of aut gamma centralizes l or l is of lie type and aut gamma involves an explicitly known same characteristic module for l c 1999 academic press

on the mumford tate group of an abelian variety with complex multiplication let k be a primitive cm type with k o 2n for definitions and previous results see section 1 1 fix n and consider the collection s n rank where rank counts the number of independent translates of qi under the galois action and k ranges over all primitive types the smallest element of s n denoted by b n is referred to as the sharp lower bound for the rank in dimension n as was brought to the author s attention by ribet bounds on b n of the form p 1 for p a prime dividing n follow directly from the proof of ribet s nondegeneracy theorem 21 this is recorded as theorem 1 4 ribet s method also gives bounds of the form 2q for q a prime with q2 dividing n as is observed in theorem 1 12 when combined with the author s constructions of abelian varieties in s 91 we obtain the precise value of b n for many values of n corollaries 1 5 to 1 8 and 1 13 we recall that these constructions use the analytic method of weil and shimura together with new results on the reflex field from an investigation suggested to the author by shim ura the interest of the rank comes from the theory of complex multiplication if a is an abelian variety of cm type k then the kubota rank of a is rank and controls properties of the classfields constructed from a as in kubota 161 and ribet 20 this is connected with the fact that the rank of is also the dimension of the mumford tate group of a and with the relation of this group to the i adic representations of a as in serre 24 251 the main body of the paper contains results that provide information on s n the main result theorem 2 5 asserts that when n is odd there is a computable subset s n of s n that accounts for the ranks of many cm types on most cm fields more precisely let k be the maximal totally real subfield of k and kg be the galois closure of k we consider the per

a geometrical method towards first integrals for dynamical systems we develop a method based on darboux s and liouville s works to find first integrals and or invariant manifolds for a physically relevant class of dynamical systems without making any assumption on these elements forms we apply it to three dynamical systems lotka volterra lorenz and rikitake

new criteria for the solvability of finite groups see 6 or 7 p 291 it has been conjectured that a group is solvable if and only if it has a hall 2 subgroup and a hall 3 subgroup see 11 we verify this conjecture for groups whose composition factors are known simple groups along the way we also verify for groups whose composition factors are known simple groups a conjecture of hall s that if a group has a hall

a new class of integrable systems we present a family of dynamical systems associated with the motion of a particle in two space dimensions these systems possess a second integral of motion quadratic in velocities apart from the hamiltonian and are thus completely integrable they were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation pde for the potential a most important point is that the same pde was derived through considerations on the analytic structure of the singularities of the solutions weak painleve property

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

painleve property and geometry abstract the painleve property for discrete hamiltonian systems implies the existence of a symplectic manifold which augments the original phase space and on which the flows exist and are analytic for all times the augmented manifold is constructed by expanding the hamilton jacobi equation a complete classification of the types of poles allowed in complex time is given for hamiltonians which separate into the direct product of hyperelliptic curves for such systems bounds on the degrees of the polynomial separating variable change and the other integrals in involution can be found from the pole series and the hamilton jacobi equation it is shown how branching can arise naturally in a painleve system

finite groups with certain subgroups of prime power order complemented abstract let p be a sylow p subgroup of a finite group g and m be a given integer satisfying 1 m 1 log p p we investigate the structure of g under the assumption that all subgroups of p with order p m are complemented in g

hall subgroups and π separable groups abstract let g be a finite group and a set of primes in this paper some new criteria for separable groups and solvable groups in terms of hall subgroups are proved the main results are the following theorem g is a separable group if and only if 1 g satisfies e and e 2 g satisfies e q and e p for all p q theorem g is a separable group if and only if 1 g satisfies e and e 2 g satisfies e p q for all p and q

finite groups with certain subgroups of sylow subgroups complemented abstract let g be a finite group and h a subgroup of g we say that h is complemented in g if there exists a subgroup k of g such that g h k and h k 1 for each prime p dividing the order of g let p be a sylow p subgroup of g we fix in each p a subgroup d such that 1 d p and study the structure of g under the assumption that each subgroup h of p with h d and h p d is complemented in g

on nonparametric maximum likelihood estimation with double truncation doubly truncated survival data arise if failure times are observed only within certain time intervals the nonparametric maximum likelihood estimator is widely used to estimate the underlying failure time distribution using a directed graph representation of the data suggested by vardi 1985 a certain graphical condition holds if and only if the nonparametric maximum likelihood estimate exists and is unique if this condition does not hold then such an estimate may exist but need not be unique so another graphical condition is proposed to check whether such an estimate exists the conditions are simple to check using existing graphical software reanalysis of an aids incubation time dataset shows that a nonparametric maximum likelihood estimate does not exist for these data

exceptional polynomials of affine type abstract let k be a finite field of characteristic p a polynomial f with coefficients in k is said to be exceptional if it induces a permutation on infinitely many finite extensions of k let t be a transcendental and k be an algebraic closure of k the exceptional polynomials known to date are constructed from certain twists of polynomials g such that the splitting field of g x t over k is rational on the other hand we know that except for p 2 or 3 either the degree of an exceptional polynomial f is a power of the characteristic p or f is a dickson polynomial the case deg f p has been settled completely we give a new series of exceptional polynomials of degree p m with m even which does not follow the above mentioned construction principle we show under certain additional hypotheses that the associated monodromy groups of exceptional polynomials will be severely restricted in particular we determine precisely what the geometric monodromy groups of exceptional polynomials of degree p 2 will be

primitive permutation groups of odd degree and an application to finite projective planes one of the most beautiful and important results concerning finite projective planes is the ostrom wagner theorem 26 such a plane admitting a 2 transitive collineation group must be desarguesian it has long been conjectured that the same conclusion must hold if it is only assumed that there is a collineation group transitive on incident point line pairs ll pp 208 214 171 the starting point for this paper was a proof of this conjecture modulo a degenerate situation

poisson structure of dynamical systems with three degrees of freedom it is shown that the poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one form in three dimensions advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability techniques for finding poisson structures are presented and applied to various examples such as the halphen system which has been studied as the two monopole problem by atiyah and hitchin it is shown that the halphen system can be formulated in terms of a flat sl 2 r valued connection and belongs to a nontrivial godbillon vey class on the other hand for the euler top and a special case of three species lotka volterra equations which are contained in the halphen system as limiting cases this structure degenerates into the form of globally integrable bi hamiltonian structures the globally integrable bi hamiltonian case is a linear and the sl 2 r structure is a quadratic unfolding of an integrable one form in 3 1 dimensions it is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of poisson structure and some new techniques for incorporating arbitrary constants into the poisson one form are presented herein this leads to some extensions analogous to q extensions of poisson structure the kermack mckendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the lorenz lotka volterra may leonard and maxwell bloch systems admit globally integrable bi hamiltonian structure

quasiprimitive groups containing a transitive alternating group abstract this paper classifies quasiprimitive permutation groups with a transitive subgroup which is isomorphic to a n for some n 5

hamiltonians with high order integrals and the weak painleve concept we examine the singularity structure of the equations of motion associated to integrable two dimensional hamiltonians with second integrals of order higher than 2 we show in these specific examples that the integrability is associated to a singularity expansion of the weak painleve type new cases of integrability are discovered with still higher order integrals which are explicitly computed

a semiparametric model for randomly truncated data abstract for randomly censored data it is known that the maximum likelihood estimate mle of the survival curve is not affected by parametric assumption on the censoring variable the kaplan meier 1958 estimate is the mle for both nonparametric and semiparametric models for randomly truncated data the truncation product limit estimate is the mle for nonparametric models this is not the case if the truncation mechanism is parameterized however specifically let x be a generic random variable and t be the truncation variable if the distribution of t is parameterized and the distribution of x is left unspecified it can be shown that the truncation product limit estimate is not the mle for this semiparametric model even though it is for the fully nonparametric model in this article the mle is characterized for the semiparametric model and the large sample properties of the estimate are established the results show that unlike censoring the parametric information from the truncation mechanism

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

on the complete symmetry group of the classical kepler system a rather strong concept of symmetry is introduced in classical mechanics in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey accordingly a complete symmetry group realization in mechanics must be endowed with the following two features 1 the group acts freely and transitively on the manifold of all allowed motions of the system 2 the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group this program is applied successfully to the classical kepler problem since the complete symmetry group for this particular system is here obtained the importance of this result for the quantum kinematic theory of the kepler system is emphasized

on fully ramified brauer characters let z be a normal subgroup of a finite group let p 5 be a prime and let ibr z be an irreducible g invariant p brauer character of z suppose that g e for some ibr g then g z is solvable in other words a twisted group algebra over an algebraically closed field of characteristic not 5 with a unique class of simple modules comes from a solvable group

lie point symmetries and first integrals the kowalevski top we show how the lie group analysis method can be used in order to obtain first integrals of any system of ordinary differential equations the method of reduction increase of order developed by nucci j math phys 37 1772 1775 1996 is essential noether s theorem is neither necessary nor considered the most striking example we present is the relationship between lie group analysis and the famous first integral of the kowalevski top

the trivial intersection problem for characters of principal indecomposable modules abstract for a prime p and a finite group g let p g denote the complex character associated to the projective indecomposable module in characteristic p with trivial head let irr p g denote the set of irreducible characters occurring as constituents in p g we characterize all finite simple groups which satisfy irr p g irr q g 1 g for all primes p q

hidden linearity in systems for competition with evolution in ecology and finance abstract in j c sprott competition with evolution in ecology and finance phys lett a 325 2004 329 333 a variant of the multispecies lotka volterra model was proposed in which species competing for fixed finite resources are replaced by a new randomly chosen species whenever they become extinct we applied lie group analysis to this system and found that it can be reduced to a linear equation if certain values are given to the parameters involved which rules out any chaotic behavior in these instances also different routes to linearization are shown

first integrals and symmetries of time dependent hamiltonian systems in this paper first integrals of one dimensional time dependent hamiltonians h q p t are studied first of all it is shown that the second first integral j q h t t dq hp valid for autonomous hamiltonians h q p can be extended to the case of nonautonomous hamiltonians h q p t provided hp resp t is replaced with ip where i q p t is a first integral of h q p t resp a new time t j q i t t dq ip in a second part we derive a technique leading to a first integral i q p t from a lie symmetry as an example this technique together with the expression of the second first integral j q i t is used to completely integrate a special case of the linear harmonic oscillator with a time dependent frequency as a result lewis first integral is recovered from a lie symmetry in a way that generalizes the recent work of shivamoggi and muilenberg phys lett a 154 1 1991

an answer to a question of isaacs on character degree graphs abstract let n be a normal subgroup of a finite group g we consider the graph g n whose vertices are the prime divisors of the degrees of the irreducible characters of g whose kernel does not contain n and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of g whose kernel does not contain n we prove that if g n is disconnected then g n is solvable this proves a strong form of a conjecture of isaacs

lorenz integrable system moves a la poinsot a transformation is derived which takes the lorenz integrable system into the well known euler equations of a torque free rigid body about a fixed point i e the famous motion a la poinsot the proof is based on lie group analysis applied to two third order ordinary differential equations admitting the same two dimensional lie symmetry algebra lie s classification of two dimensional symmetry algebras in the plane is used if the same transformation is applied to the lorenz system with any values of the parameters then one obtains euler equations of a rigid body about a fixed point subjected to a torsion depending on time and angular velocity the numerical solution of this system yields a three dimensional picture which looks like a tornado the cross section of which has a butterfly shape thus lorenz s butterfly has been transformed into a tornado

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

on the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems in this paper we first give a necessary condition for general nonlinear systems to have rational first integrals then by using the so called kowalevsky exponents we present a criterion for nonexistence of rational first integrals for semiquasihomogeneous systems

finitely generated groups p adic analytic groups and poincare series this is equivalent to the coefficients apn g satisfing a linear recurrence relation with constant coefficients for sufficiently large n these poincare series were first studied in the case when g is a finitely generated torsion free nilpotent group in a paper by grunewald segal and smith gssm there the authors established that for such a group g p s is a rational function in p8s in that setting the functions g p s are the local factors associated with the dirichlet series

on c supplemented subgroups of finite groups abstract let g be a finite group a subgroup h of g is said to be c supplemented in g if there exists a subgroup k of g such that g h k and h k h g where h g g g h g is the largest normal subgroup of g contained in h we investigate the structure of g under the assumption that some families of subgroups of g are c supplemented in g

the complete kepler group can be derived by lie group analysis it is shown that the complete symmetry group for the kepler problem as introduced by krause can be derived by lie group analysis the same result is true for any autonomous system

integrating the nonintegrable analytic structure of the lorenz system revisited abstract a study of the complex time analytic structure of the lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one time dependent integral of motion exists furthermore the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument the analytic techniques used in these studies a systematic resummation of a logarithmic psi series appears to be quite general and can provide explicit representations of a solution even in the chaotic regimes in the neighborhood of a given movable singularity furthermore we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

dynamical systems embedded into lie algebras analytical and geometrical information on certain dynamical systems x is obtained under the assumption that x is embedded into a certain real lie algebra

lie symmetries and invariants of the lotka volterra system in this paper we use the lie symmetry method for finding rational and transcendental symmetry transformations and invariants for the 3d lotka volterra system

necessary conditions for the existence of quasi polynomial invariants the quasi polynomial and lotka volterra systems abstract we show that any quasi polynomial invariant of a quasi polynomial dynamical system can be transformed into a quasi polynomial invariant of a homogeneous quadratic lotka volterra dynamical system we show how this quasi polynomial invariant can be decomposed in a simple manner this decomposition permits to conclude that the existence of polynomial semi invariants in lotka volterra systems is a necessary condition for the existence of quasi polynomial invariants we derive a method which allows to construct the necessary conditions for existence of semi invariants on lotka volterra dynamical systems applications are given

the determination of nonlocal symmetries by the technique of reduction of order autonomous systems of ordinary differential equations can be rewritten as systems of first order ordinary differential equations and one of the dependent variables chosen as a new independent variable some of the variables are eliminated to give a mixed system of first and second order equations for which the determination of point symmetries can be automated without having to make an ansatz on the detailed structure of the symmetry because the coefficient function for the original independent variable appears only as its derivative in the reduced system symmetries which are nonlocal in this variable become local symmetries of the reduced system and can be computed algorithmically

lie symmetries for the reduced three wave interaction problem the authors show how the method of lie symmetries can be used to obtain first integrals and for the identification of completely or partially integrable cases of the reduced three wave interaction problem

lax representation for the systems of s kowalevskaya type we describe a number of dynamical systems that are generalizations of the s kowalevskaya system and admit the lax representation

calogero s goldfish is indeed a school of free particles a many body system of n nonlinear ordinary differential equations of second order which is amenable to exact treatments a goldfish calogero 2001 the neatest many body problem amenable to exact treatments a goldfish physica d 152 153 78 84 is shown to be equivalent through an exact transformation to the equations of one dimensional motion of n 1 free particles a school of free particles indeed the transformation is obtained by applying the reduction method and lie group analysis as introduced in nucci 1996 the complete kepler group can be derived by lie group analysis j math phys 37 1772 5

groups whose prime graphs have no triangles abstract let g be a finite group and let cd g be the set of all complex irreducible character degrees of g let g be the set of all primes which divide some character degree of g the prime graph g attached to g is a graph whose vertex set is g and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of g in this paper we show that if g is a finite group whose prime graph g has no triangles then g has at most 5 vertices we also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups finally we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices

on fφ hypercentral subgroups of finite groups let g be a finite group we write r g to denote the largest soluble normal subgroup of g and put g r g we say that a chief factor h k of g is non frattini non solubly frattini if h k g k if h k g k respectively a chief factor h k of g is called f central in g provided h k g cg h k f a normal subgroup n of g is said to be f hypercentral f hypercentral in g if either n 1 or n 1 and there exists a chief series 1 n0 n1 nt n of g below n such that every non frattini non solubly frattini respectively factor ni ni 1 of series is f central in g in this paper we analyze some properties and applications of f hypercentral and f hypercentral subgroups

nonparametric methods for doubly truncated data abstract truncated data play an important role in the statistical analysis of astronomical observations as well as in survival analysis the motivating example for this article concerns a set of measurements on quasars in which there is double truncation that is the quasars are observed only if their luminosity occurs within a certain finite interval bounded at both ends with the interval varying for different observations nonparametric methods for the testing and estimation of doubly truncated data are developed these methods extend some known techniques for data that are truncated only on one side in particular lynden bell s estimator and the truncated version of kendall s tau statistic however the kind of hazard function arguments that underlie the one sided methods fail for two sided truncation bootstrap and markov chain monte carlo techniques are used here in their place finally we apply these techniques to the quasar data answering a question about their long term luminosity evolution

large sample theory of empirical distributions in biased sampling models vardi 1985a introduced an s sample model for biased sampling gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator mathbb g n of the common underlying distribution g and discussed numerical methods for calculating the estimator here we examine the large sample behavior of the npmle mathbb g n including results on uniform consistency of mathbb g n convergence of sqrt n mathbb g n g to a gaussian process and asymptotic efficiency of mathbb g n as an estimator of g the proofs are based upon recent results for empirical processes indexed by sets and functions and convexity arguments we also give a careful proof of identifiability of the underlying distribution g under connectedness of a certain graph mathbf g examples and applications include length biased sampling stratified sampling enriched stratified sampling choice based sampling in econometrics and case control studies in biostatistics a final section discusses design issues and further problems

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

the probabilistic zeta function of finite simple groups abstract let g be a finite group there exists a uniquely determined dirichlet polynomial p g s such that if t n then p g t gives the probability of generating g with t randomly chosen elements we show that it may be recognized from the knowledge of p g s whether g frat g is a simple group

two results related to the solvability of m groups abstract the results of this paper are as follows for a finite group g if all primitive irreducible characters are linear or if all nonmonomial irreducible characters have the same degree then g is solvable

on finite factorizable groups in this paper we use freely the recent classification theorem for finite simple groups from the early stages of finite group theory finite factorizable groups satisfying h were studied by many authors in 1904 burnside 6 proved the solvability of groups factorizable by two prime power subgroups many years later wielandt 29 proved the solvability of finite groups factorizable by two nilpotent subgroups of relatively prime orders kegel extended wielandt s result in the case that both subgroups a and b are nilpotent without restrictions the second author in 121 classified finite factorizable groups of type h where a and b are solvable subgroups results on groups satisfying h can be found in 3 4 10 12 19 20 291 and in other papers on finite factorizable groups theorem 1 1 and corollary 1 2 of this paper state the final solution to the problem of classifying groups of type h we prove the following

algebraic structures and invariant manifolds of differential systems algebraic tools are applied to find integrability properties of odes bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations since all equations in this class are related to a canonical quadratic differential system the lotka volterra system these algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given their subalgebras are associated to nonlinear invariant manifolds in the phase space these manifolds are calculated explicitly more general algebraic invariant surfaces are also obtained by combining a theorem of walcher and the lotka volterra canonical form applications are given for lorenz model lotka may leonard and rikitake systems

restrictions on the structure of subgroup lattices of finite alternating and symmetric groups let g be a finite alternating or symmetric group we describe an infinite class of finite lattices none of which is isomorphic to any interval h g in the subgroup lattice of g

lie symmetries and infinite dimensional lie algebras of certain nonlinear dissipative systems we point out that the dissipative force free duffing oscillator and holmes rand nonlinear oscillator admit infinite dimensional symmetry algebras for certain parametric choices for which the systems become integrable from the nature of the symmetry vector fields one can also write down the integrals of motion for the above systems in addition we point out that the other dissipative systems discussed in the literature also admit infinite dimensional lie algebras

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

on primitive overgroups of quasiprimitive permutation groups a permutation group is said to be quasiprimitive if all its non trivial normal subgroups are transitive we investigate pairs g h of permutation groups of degree n such that g h sn with g quasiprimitive and h primitive an explicit classification of such pairs is obtained except in the cases where the primitive group h is either almost simple or the blow up of an almost simple group the theory in these remaining cases is investigated in separate papers the results depend on the finite simple group classification

on weakly s permutable subgroups of finite groups abstract let g be a finite group h a subgroup of g and h s g the subgroup of h generated by all those subgroups of h which are s permutable in g then we say that h is weakly s permutable in g if g has a subnormal subgroup t such that h t g and t h h s g we fix in every non cyclic sylow subgroup p of g a subgroup d satisfying 1 d p and study the structure of g under the assumption that all subgroups h with h d are weakly s permutable in g

on a class of non solvable groups abstract in this paper we use the properties of subgroups with given order to study the structure of finite groups the main result is as follows let g be a group and p be a sylow p subgroup of g suppose that 1 d p if every subgroup h of p with h d is m supplemented in g then every non abelian pd g chief factor a b satisfies one of the following conditions 1 a b p s l 2 7 and p 7 a b p s l 2 11 and p 11 2 a b p s l 2 2 t and p 2 t 1 3 is a fermat prime 3 a b p s l n q n 3 is a prime n q 1 1 and p q n 1 q 1 4 a b m 11 and p 11 a b m 23 and p 23 5 a b a p and p 5

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory

a lax pair for kowasevski s top abstract we show that on each level surface of the invariants the equations of the kowalevski top are equivalent to a neumann system describing the motion of a mass point on the sphere s2 p 1 under the influence of force qp this allows us to write a global lax pair for the kowalevski system and to show that kowalevski s original reduction of the integration of the equations of motion to jacobi s inversion theorem is identical to the introduction of elliptical spherical coordinates for integrating c neumann s system

kowalewski s top on the lie algebras o 4 e 3 and o 3 1 the authors extend the lax pair for kowalewski s top kt obtained by haine and horozov 1987 to the lie algebras o 4 and o 3 1 the results are expressed in terms of a solution of the neumann system they derive formulae for the action variables by analogy with the e 3 kt case and obtain a useful representation for the equations of motion that demonstrate explicitly the separation of variables