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

inflection points and topology of surfaces in 4 space we consider asymptotic line fields on generic surfaces in 4 space and show that they are globally defined on locally convex surfaces and their singularities are the inflection points of the surface as a consequence of the generalized poincare hopf formula we obtain some relations between the number of inflection points in a generic surface and its euler number in particular it follows that any 2 sphere generically embedded as a locally convex surface in 4 space has at least 4 inflection points

on representing finite lattices as intervals in subgroup lattices of finite groups let mnbe the lattice of length 2 withn 1 atoms it is an open problem to decide whether or not every such lattice or indeed whether or not every finite lattice can be represented as an interval in the subgroup lattice of some finite group we complete the work of the second author lucchini by reducing this problem to a series of questions concerning the finite non abelian simple groups

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

ovaloids of r3 and their umbilics a differential equation approach abstract the following extension of the caratheodory conjecture on umbilics is proposed and solved in very special cases let n m s 2 be the gauss map of a c 3 ovaloid m r 3 let a be the set of umbilics q m such that n q q if a is a connected component of a then each connected component of m a contains at least an umbilic point of m

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

the mckay conjecture and galois automorphisms the main problem of representation theory of finite groups is to find proofs of several conjectures stating that certain global invariants of a finite group g can be computed locally the simplest of these conjectures is the mckay conjecture which asserts that the number of irreducible complex characters of g of degree not divisible by p is the same if computed in a p sylow normalizer of g in this paper we propose a much stronger version of this conjecture which deals with galois automorphisms in fact the same idea can be applied to the celebrated alperin and dade conjectures

blocks with cyclic defect groups and clifford extensions abstract we prove that the residual clifford extension of any non exceptional character belonging to a p block with cyclic defect groups of a twisted group algebra is naturally isomorphic to the clifford extension of any simple module belonging to that block we also show that dade s inductive conjecture holds for blocks with cyclic defect groups

the mckay conjecture is true for the sporadic simple groups abstract the mckay conjecture states that the number of irreducible complex characters of a group g that have degree prime to p is equal to the same number for the sylow p normalizer in g we verify this conjecture for the 26 sporadic simple groups

on binary differential equations and umbilics in this paper we give the local classification of solution curves of bivalued direction fields determined by the equationwhere a and b are smooth functions which we suppose vanish at 0 2 such fields arise on surfaces in euclidean space near umbilics as the principal direction fields and also in applications of singularity theory to the structure of flow fields and monochromatic electromagnetic radiation we give a classification up to homeomorphism there are three types but the methods furnish much additional information concerning the fields via a crucial blowing up construction

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

dade s conjecture for the simple o nan group let g be the simple o nan group it has order 29 34 5 73 11 19 31 it has schur multiplier of order 3 and aut g g is cyclic of order 2 in this paper we prove that dade s conjecture for g is true here we mean by saying the conjecture the inductive form of the conjecture see 3 according to 3 the inductive form is equivalent to the invariant projective form if the outer automorphism group has cyclic sylow q subgroups for each prime q thus for the simple o nan group it suffices to check the invariant projective form moreover if a defect group of a block is cyclic then the invariant projective form is proved to be true for this block in 4 thus it suffices to treat the primes 2 3 and 7 in this paper a b and a b denote a split and a nonsplit extension of a by b respectively we use n and dn to denote a cyclic and a dihedral group of order n respectively moreover for a prime p an elementary abelian group of order p is denoted by p for an odd prime p we denote by p1 2n an extraspecial group of order p1 2n and exponent p and by 21 2n a minus type extraspecial group of order 21 2n finally sn and an denote the symmetric group and the alternating group of degree n respectively

convergence and coupling for spin glasses and hard spheres we discuss convergence and coupling of markov chains and present general relations between the transfer matrices describing these two processes we then analyze a recently developed local patch algorithm which computes rigorous upper bound for the coupling time of a markov chain for nontrivial statistical mechanics models using the coupling from the past protocol this allows one to exactly sample the underlying equilibrium distribution for spin glasses in two and three spatial dimensions the local patch algorithm works at lower temperatures than previous exact sampling methods we discuss variants of the algorithm which might allow one to reach in three dimensions the spin glass transition temperature the algorithm can be adapted to hard sphere models for two dimensional hard disks the algorithm allows us to draw exact samples at higher densities than previously possible

verification of dade s conjecture for janko groupj3 abstract in 3 dade made a conjecture expressing the number k b d of characters of a given defect d in a given p block b of a finite group g in terms of the corresponding numbers k b d for blocks b of certain p local subgroups of g several different forms of this conjecture are given in 5 dade claims that the most complicated form of this conjecture called the inductive conjecture 5 8 in 5 will hold for all finite groups if it holds for all covering groups of finite simple groups in this paper we verify the inductive conjecture for all covering groups of the third janko group j 3 in the notation of the atlas 1 this is one step in the inductive proof of the conjecture for all finite groups certain properties of j 3 simplify our task the schur multiplier of j 3 is cyclic of order 3 see 1 p 82 hence there are just two covering groups of j 3 namely j 3 itself and a central extension 3 j 3 of j 3 by a cyclic group z of order 3 we treat these two covering groups separately the outer automorphism group out j 3 of j 3 is cyclic of order 2 see 1 p 82 in this case dade affirms in 5 section 6 that the inductive conjecture for j 3 is equivalent to the much weaker invariant conjecture 2 5 in 5 furthermore dade has proved in 6 that this invariant conjecture holds for all blocks with cyclic defect groups the sylow p subgroups of j 3 are cyclic of order p for all primes dividing j 3 except 2 and 3 so we only need to verify the invariant conjecture for the two primes p 2 and p 3 we do that in theorem 2 10 1 and theorem 3 6 1 below the group out 3 j 3 z of outer automorphisms of 3 j 3 centralizing z is trivial see 1 p 82 in this case dade affirms that the inductive conjecture is equivalent to the projective conjecture 4 4 in 5 again dade has shown in 6 that this projective conjecture holds for all blocks with cyclic defect groups so we only need to verify the projective conjecture for p 2 and p 3 we do that in theorem 4 4 2 and theorem 4 3 1 for the above reasons the four theorems 2 10 1 3 6 1 4 4 2 and 4 3 1 below are sufficient to prove the inductive form of dade s conjecture for all covering groups of j 3 this paper is my ph d thesis it would not have been possible without the help and infinite patience of my advisor everett dade

uno s invariant conjecture for the general linear and unitary groups in nondefining characteristics abstract uno s invariant conjecture which is a combination of dade s invariant conjecture and the isaacs navarro conjecture is proved for general linear and unitary groups in nondefining characteristics

positive quadratic differential forms and foliations with singularities on surfaces to every positive cr quadratic differential form defined on an oriented two manifold is associated a pair of transversal one dimensional crfoliations with common singularities an open set of positive cr quadratic differential forms with structural stable associated foliations is characterized and it is proved that this set is dense in the space of positive c quadratic differential forms with c2 topology also a realization theorem is established

a reduction theorem for the alperin weight conjecture the alperin weight conjecture was stated by j alperin in 1986 and constitutes one of the main problems in the representation theory of finite groups in this paper we provide a reduction to simple groups and as an application we prove the conjecture in several cases

renormalization group approach to exact sampling in this letter we use a general renormalization group algorithm to implement propp and wilson s coupling from the past approach to complex physical systems our algorithm follows the evolution of the entire configuration space under the markov chain monte carlo dynamics from parts of the configurations patches on increasing length scales and it allows us to generate exact samples of the boltzmann distribution which are rigorously proven to be uncorrelated with the initial condition we validate our approach in the two dimensional ising spin glass on lattices of size 64 ifmmode times else texttimes fi 64

dade s invariant conjecture for steinberg s triality groups d43 2n in defining characteristic abstract in this paper we verify dade s invariant conjecture for steinberg s triality groups d 4 3 2 n in the defining characteristic i e in characteristic 2 together with the results in j an dade s conjecture for steinberg triality groups d 4 3 q in non defining characteristics math z 241 2002 445 469 and j an f himstedt s huang uno s invariant conjecture for steinberg s triality groups in defining characteristic in preparation this completes the proof of dade s conjecture for steinberg s triality groups furthermore we show that the isaacs malle navarro version of the mckay conjecture holds for d 4 3 2 n in the defining characteristic i e d 4 3 2 n is good for the prime 2 in the sense of isaacs malle and navarro

quantum algorithm for exact monte carlo sampling we build a quantum algorithm which uses the grover quantum search procedure in order to sample the exact equilibrium distribution of a wide range of classical statistical mechanics systems the algorithm is based on recently developed exact monte carlo sampling methods and yields a polynomial gain compared to classical procedures

uno s invariant conjecture for steinberg s triality groups in defining characteristic abstract we verify uno s invariant conjecture for steinberg s triality groups d 4 3 q q a power of an odd prime p in the defining characteristic p uno s invariant conjecture is a refinement of dade s invariant conjecture

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

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

dade s invariant conjecture for general linear and unitary groups in non defining characteristics this paper is part of a program to study the conjecture of e c dade on counting characters in blocks for several finite groups the invariant conjecture of dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups