groups with just one character degree divisible by a given prime the ito michler theorem asserts that if no irreducible character of a finite group g has degree divisible by some given prime p then a sylow p subgroup of g is both normal and abelian in this paper we relax the hypothesis and we assume that there is at exactly one multiple of p that occurs as the degree of an irreducible character of g we show that in this situation a sylow p subgroup of g is almost normal in g and it is almost abelian

on heights of characters of finite groups abstract in this paper we first state a conjecture on the lower bound of the maximal height of characters in a p block of a finite group then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group for all blocks of a quasi simple group g with g z g isomorphic to a 6 a 7 or a simple group of lie type with an exceptional covering group for all blocks of a symmetric group as well as for all blocks of finite general linear or unitary groups for the proof of the case with symmetric groups it relates to an open question of olsson on the existence of t core partitions where t 3 is an integer as a byproduct our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the isaacs moreto navarro tiep conjecture claiming that the number of distinct irreducible character degrees of a sylow p subgroup p of an arbitrary finite group g is at most one more than the number of irreducible character degrees of g that are multiples of p

brauer s height zero conjecture for quasi simple groups abstract we show that brauer s height zero conjecture holds for blocks of finite quasi simple groups this result is used in navarro spath s reduction of this conjecture for general groups to the inductive alperin mckay condition for simple groups

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

melnikov transforms bernoulli bundles and almost periodic perturbations in this paper we study nonlinear time varying perturbations of an autonomous vector field in the plane r2 we assume that the unperturbed equation i e the given vector field has a homoclinic orbit and we present a gen eralization of the melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory the key to our generalization is the concept of the melnikov transform which is a linear transformation on the space of perturbation functions the appropriate dynamical setting for studying these perturbation is the concept of a skew product flow the concept of transversality we require is best understood in this context under conditions whereby the perturbed equation admits a transversal homoclinic trajectory we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow we show the dynamics near this trajectory can have the exotic behavior of the bernoulli shift the exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle which we call the bernoulli bundle we allow all perturbations which are bounded and uniformly continuous in time thus our theory includes the classical periodic perturbations studied by melnikov quasi periodic and almost periodic perturbations as well as toroidal perturbations which are close to quasi periodic perturbations

cantori for multiharmonic maps abstract we compute all the cantori and their gap and turnstile structures for area preserving twist maps near non degenerate anti integrable limits with arbitrarily many wells per period the results imply a rich bifurcation diagram for cantori of families of maps containing degenerate anti integrable limits we conjecture the structure of this bifurcation diagram in the case of the two harmonic family

resonant overlap phenomena in stochastic layers of non linear hamiltonian systems with periodical excitations an accurate standard map approach instead of the approximate standard map approach is developed for the analytical prediction of the onset of a new resonant overlap forming a new resonant separatrix web in the stochastic layer this accurate approach applies to the twin well duffing oscillator for illustration of the resonant overlap phenomena in stochastic layers the excitation strengths are obtained analytically for the onset of the specified resonant separatrix webs in the stochastic layer this accurate approach gives a very good prediction compared to numerical results for the lower order resonant overlap when excitation strength is very weak the accurate and approximate standard map methods are in good agreement and the accurate one is applicable to non linear systems with strong excitations however the further improvement should be completed for a more accurate analytical prediction of the onset of a new resonant overlap in stochastic layers of non linear hamiltonian systems with periodical excitations

some congruences for partitions that are p cores a number of linear congruences modulo r are proved for the number of partitions that are p cores where p is prime 5 p 23 and r is any prime divisor of 1 2 p 1 analogous results are derived for the number of irreducible p modular representations of the symmetric group s n the congruences are proved using the theory of modular forms

canonical maps near separatrix in hamiltonian systems a systematic and rigorous method to construct symplectic maps near separatrix of generic hamiltonian systems subjected to time periodic perturbations is developed it is based on the method of canonical transformation of variables to construct hamiltonian maps j phys a 35 2811 2002 using canonical transformation of variables and the first order approximation for the generating function the general form of mapping in terms of time and energy variables is obtained different limiting cases of the mapping are considered the method is illustrated for simple hamiltonian systems with one and a large number of saddle points it is also applied to derive mappings for the periodic driven morse oscillator describing the process of stochastic excitation and dissociation of diatomic molecules the so called canonical kepler map is derived for the one dimensional hydrogen atom in a microwave field

width of stochastic layers in near integrable two dimensional symplectic maps abstract width of stochastic layers around separatrix eight is studied for two dimensional symplectic maps close to integrable we prove that under rather general assumptions the relation w d 1 log holds where w is the width of the stochastic layer d is the width of a lobe domain and is a multiplier at the corresponding hyperbolic point

a note on the thin finite simple groups in a previous paper 11 the author described a new approach to nonabelian simple groups of characteristic 2 type in which every 2 local subgroup of odd index is solvable and has cyclic sylow subgroups for all odd primes in fact we imposed these conditions only on certain special 2 local subgroups of odd index the approach was made possible by certain new general results concerning groups of characteristic 2 type which divide into the following three categories

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

phase change between separatrix crossings abstract the change of the crossing parameter essentially the phase between separatrix crossings is calculated for hamiltonian systems with one degree of freedom and slow time dependence this completes the calculation of the map for an arbitrary sequence of separatrix crossings the change of the crossing parameter is used to calculate the retrapping probability it is found that even in the limit of infinitely slow time dependence correlations persist between the separatrix crossings and corrections to the usual zero order adiabatic approximation must be included comparison with numerical experiments show that the result derived here is accurate for quite large values of the slowness parameter

analysis of the separatrix map in hamiltonian systems abstract a one degree of freedom hamiltonian system with time periodic perturbation generally display rich dynamics near the separatrix the understanding of which is important in studying transport and diffusion we introduce the exact separatrix map which provides an effective way to describe dynamics near the separatrix we show that the separatrix map and its inverse satisfy twist conditions with logarithmic singularities and therefore they have an infinite twist near the transversal intersections of the stable and unstable manifolds of the hyperbolic equilibrium using the variational formalism with the generating functions we show the dynamics near the singularities correspond to the anti integrable limit as a consequence we show the nonexistence of rotational invariant circles with sufficiently large rotation number first obtained by lazutkin in standard maps moreover the existence of a uniformly hyperbolic chaotic invariant set is shown in connection with the theory of the anti integrable limit the simplest form of the separatrix map is studied in detail to illustrate some of the above results explicitly

chaotic trajectories in the standard map the concept of anti integrability a rigorous proof is given in the standard map associated with a frenkel kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k k0 these chaotic trajectories with finite entropy per site are coded by integer sequences mi such that the sequence bi mi 1 mi 1 2mi be bounded by some integer b the bound k0 in k depends on b and can be lowered for coding sequences mi fulfilling more restrictive conditions the obtained chaotic trajectories correspond to stationary configurations of the frenkel kontorowa model with a finite non zero photon gap called gap parameter in dimensionless units this property implies that the trajectory or the configuration ui can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration a non zero gap parameter implies that the lyapunov coefficient is strictly positive when it is defined in addition the existence of dilating and contracting manifolds is proven for these chaotic trajectories exotic trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems the concept of anti integrability emerges from these theorems in the anti integrable limit which can be only defined for a discrete time dynamical system the coordinates of the trajectory at time i do not depend on the coordinates at time i 1 thus at this singular limit the existence of chaotic trajectories is trivial and the dynamical system reduces to a bernoulli shift it is well known that the kam tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit under certain conditions in a similar way it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti integrable limits also under certain conditions

character correspondences in finite general linear unitary and symmetric groups let r 0 be a prime integer and let g be a finite general linear group gl n q a unitary group u n q with q qg or a symmetric group s n of degree n the partition of the irreducible ordinary characters of s n into r blocks of s n was given by brauer s and robinson s solution of nakayama s conjecture see 7 p 245 in their fundamental paper 5 fong and srinivasan have recently classified the r blocks of gl n q and u n q for all primes r 2 with r q 1 using these classifications of the r blocks of g we show in this article that there is a natural one to one correspondence b between the irreducible characters of height zero of an r block b of g with defect group r and the irreducible characters of height zero of the brauer correspondent b of b in n ng r theorem 4 10 in particular ko b ko b where ko b denotes the number of all irreducible characters of b with height h t z 0 therefore alperin s conjecture on the numbers of irreducible characters of height zero is verified for all general linear unitary and symmetric groups if g s n then theorem 4 10 also holds for r 2 in order to establish the character correspondence three other correspondences are studied the product of which is b in sect 1 we construct for every r block b of g with defect group r a subgroup g of g with a sylow r subgroup r such that there is a natural height preserving one to one correspondence t between the set irr b of all irreducible characters of b and the set irr 0 of all irreducible characters of the principal r block o of d reduction theorems 1 9 and 1 10 the map respects the geometric conjugacy classes of characters if 0 denotes the principal r block of n nd and if b is the brauer correspondent of b in n n6 r then by theorems 3 8 and 3 10 the block ideals o and b are morita equivalent have the same decomposition numbers and there is a natural height preserving one to one correspondence a between irr bo and irr b

equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for frenkel kontorova models there is a well known correspondence between the dynamics of symplectic twist maps which represent an important class of hamiltonian systems and the equilibrium states of a class of variational problems in solid state physics known as frenkel kontorova models in this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent this allows one to transfer many ideas between the two and hence to deduce new results for example we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so called non degenerate anti integrable limits by aubry and abramovici using the concept of phonon gap and deduce that they have measure zero we prove existence of many multi defect equilibrium states for a frenkel kontorova system if it has a single non degenerate defect equilibrium state via standard results in the theory of uniform hyperbolicity

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system