anatomy of the monster ii abstract we summarise much of what is currently known about the subgroup structure of the fischer griess monster introduction the publication of the atlas of finite groups 1 henceforth referred to as the atlas marked the end of a chapter in the history of group theory in more ways than one the classification of finite simple groups had been completed and using new computational techniques the last of the sporadic groups had been constructed and enough progress had been made in classifying their maximal subgroups to make the rest seem like a tidying up job since the publication of the atlas these computational methods have been pushed further the classification of the maximal subgroups for fi 22 th fi 23 j 4 and fi 24 was completed by kleidman linton parker and wilson in 7 8 9 10 12 13 14 this means that appendix 2 of the atlas of brauer characters 11 when collated with the atlas itself completes the listing for all sporadic groups except the baby monster and monster for the baby monster four new maximal subgroups s 6 l 3 4 2 2 s 6 s 6 4 l 2 31 and l 2 49 2 3 were given in appendix 2 of 11 and wilson has since found four more m 11 l 3 3 l 2 17 2 and l 2 ll 2 for references see 23 24 25 with work on the 2 locals by meierfrankenfeld and shpektorov 15 an end to the baby monster problem is in sight with a lot of work also having been done on maximal subgroups of generic groups this leaves only the monster unfortunately the degree of its smallest representation over any field 196882 is too big for present day computational techniques to make any headway

analytic and gevrey hypoellipticity for a class of pseudodifferential operators in one variable abstract we study the hypoellipticity of pseudo differential operators in one variable when a positivity assumption is made on a suitable principal part we prove that the found gevrey regularity is optimal

spectral and computational properties of band symmetric toeplitz matrices abstract we are investigating spectral properties of band symmetric toeplitz matrices bst matrices by giving a suitable representation of a bst matrix we achieve separation results and multiplicity conditions for the eigenvalues of a 7 r 5 diagonal bst matrix and also structural properties of the eigenvectors we give eigenvalue bounds for a 2k 1 diagonal bst matrix and also necessary and sufficient conditions for positive definiteness which are easy to check the same conditions apply in the case of block bst matrices either with full blocks or with bst blocks we exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a bst matrix either for sequential or for parallel computations two algorithms based on the bisection technique and newton s method are shown to be very fast for computing the eigenvalues of a 7 or 5 diagonal bst matrix

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

on beauville structures on the groups sn and an it is known that the symmetric group s n for n 5 and the alternating group a n for large n admit a beauville structure in this paper we prove that a n admits a beauville resp strongly real beauville structure if and only if n 6 resp n 7 we also show that s n admits a strongly real beauville structure for n 5

linear groups with orders having certain large prime divisors in this paper we obtain a classification of those subgroups of the finite general linear group gld q with orders divisible by a primitive prime divisor of qe 1 for some in the course of the analysis we obtain new results on modular representations of finite almost simple groups in particular in the last section we obtain substantial extensions of the results of landazuri and seitz on small cross characteristic representations of some of the finite classical groups 1991 mathematics subject classification primary 20g40 secondary 20c20 20c33 20c34 20e99

fuchsian triangle groups and grothendieck dessins variations on a theme of belyi according to a theorem of belyi a smooth projective algebraic curve is defined over a number field if and only if there exists a non constant element of its function field ramified only over 0 1 and the existence of such a belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the poincare upper half plane by a subgroup of finite index in a fuchsian triangle group on the other hand fuchsian triangle groups arise in many contexts such as in the theory of hypergeometric functions and certain triangular billiard problems which would appear at first sight to have no relation to the galois problems that motivated the above discovery of belyi in this note we review several results related to belyi s theorem and we develop certain aspects giving examples for preliminary accounts see the preprint wo1 the conference proceedings article bauitz and the comptes rendus note cowo2

fibred surfaces varieties isogenous to a product and related moduli spaces a fibration of an algebraic surface s over a curve b with fibres of genus at least 2 has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group g a variety isogenous to a higher product is the quotient of a product of curves of genus at least 2 by the free action of a finite group theorem b gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the euler number theorem c classifies the groups thus occurring and shows that after fixing the group and the euler number one obtains an irreducible moduli space the result of theorem b is extended to higher dimension in theorem g thus generalizing cf also theorem h results of jost yau and mok concerning varieties whose universal cover is a polydisk theorem a shows that fibrations where the fibre genus and the genus of the base b are at least 2 are invariants of the oriented differentiable structure the main theorems d and e characterize surfaces carrying constant moduli fibrations as surfaces having a zariski open set satisfying certain topological conditions e g having the right euler number the right fundamental group and the right fundamental group at infinity 0 introduction the study of fibrations of a smooth algebraic surface s over a smooth algebraic curve b lies at the heart of the classification theory and of the geometry of algebraic surfaces our main results in this paper theo rems d e concern a topological characterization of surfaces admitting a constant moduli fibration these results are intimately related to a series of variations on the same theme also in higher dimension and to a circle of allied techniques which we present in the course of the paper the first new result we have in this direction and which is presented in section 2 after we recall several known results some classical some quite recent is the following theorem on the dif ferentiable invariance of the genus of fibres cf theorem 2 9 for a more precise statement theorem a let f s b be a fibration of the surface s over a curve b of genus b 2 and with fibre f of genus g 1 assume that s is another smooth compact kahler surface which is orientedly diffeomorphic to s then the subspace of h 1 s corresponding to f h 1 b determines a fibration f s b with

a kronecker product variant of the facr method for solving the generalized poisson equation we present a fast direct method for the solution of a linear system mx y where m is a block tridiagonal toeplitzmatrix with a on the diagonal and t on the two subdiagonals a and t commute such matrices are obtained from a finite difference approximation to poisson s equation with nonconstant coefficients in one direction among others the new method is called kpcr l method and begins with l steps of cyclic reduction after which the remaining system is solved by a kronecker product method for an appropriate choice of l the asymptotic operation count for an n n grid is o n2 log2 log2 n which is faster than either cyclic reduction or the kronecker product method itself the algorithm is similar to and has the same complexity as the facr l algorithm which is a combination of cyclic reduction and fourier analysis or matrix decomposition however the facr l algorithm only reaches this complexity if a and t can be diagonalized by a fast transformation where the new method is fast for every banded a and t moreover the kpcr l method can be easily generalized to the case where a and t do not commute

the analysis of partial differential operators i hormander springer 1985 vol 1 391 pp vol 2 390 pp vol 3 525 pp vol 4 351 pp

the maximal subgroups of the baby monster i abstract in this paper we describe the completion of the determination of the maximal subgroups of the baby monster simple group our results are proved using computer calculations with matrix generators for the group the full details of the calculations will appear elsewhere

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

hypoellipticity and higher order levi conditions abstract we study the c hypoellipticity for a class of double characteristic operators with symplectic characteristic manifold in the case the classical condition of minimal loss of derivatives is violated

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

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

base sizes for simple groups and a conjecture of cameron let g be a permutation group on a finite set a base for g is a subset b with pointwise stabilizer in g that is trivial we write b g for the smallest size of a base for g in this paper we prove that b g 6 if g is an almost simple group of exceptional lie type and is a primitive faithful g set an important consequence of this result when combined with other recent work is that b g 7 for any almost simple group g in a non standard action proving a conjecture of cameron the proof is probabilistic and uses bounds on fixed point ratios

a special class of structured matrices constructed with the kronecker product and its use for difference equations abstract starting from a sequence of specific orthogonal matrices and using the matricial kronecker product we build classes of matrices sk k 1 2 which are shown to possess a certain affiliation with toeplitz and hankel matrices being completely determined by the first row elements their eigenvalues as well as an inversion formula are easily derived therefrom s matrices as they are termed in general may arise through the application of finite difference methods to boundary value problems

unipotent classes in maximal subgroups of exceptional algebraic groups abstract let g be a simple algebraic group of exceptional type over an algebraically closed field and m be a maximal closed connected reductive subgroup of g for each unipotent m class we determine the g class in which it lies

generation of finite quasisimple groups with an application to groups acting on beauville surfaces we verify a conjecture of bauer catanese and grunewald by proving a stronger result concerning quasisimple groups more specifically we prove that every finite quasisimple group apart from a5 and its cover sl 2 5 is a beauville group

on base sizes for almost simple primitive groups abstract let g sym be a finite almost simple primitive permutation group with socle g 0 a subset of is a base for g if its pointwise stabilizer is trivial the base size of g denoted b g is the minimal size of a base we say that g is standard if g 0 a n and is an orbit of subsets or partitions of 1 n or if g 0 is a classical group and is an orbit of subspaces or pairs of subspaces of the natural module for g 0 the base size of a standard group can be arbitrarily large in general whereas the situation for non standard groups is rather more restricted indeed we have b g 7 for every non standard group g with equality if and only if g is the mathieu group m 24 in its natural action on 24 points in this paper we extend this result by classifying the non standard groups with b g 6 the main tools include recent work on bases for actions of simple algebraic groups together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of lie type

permutation groups and binary self orthogonal codes abstract let g be a permutation group on an n element set we study the binary code c g defined as the dual code of the code spanned by the sets of fixed points of involutions of g we show that any g invariant self orthogonal code of length n is contained in c g many self orthogonal codes related to sporadic simple groups including the extended golay code are obtained as c g some new self dual codes invariant under sporadic almost simple groups are constructed

polytopes of high rank for the symmetric groups abstract in the atlas of abstract regular polytopes for small almost simple groups by leemans and vauthier the polytopes whose automorphism group is a symmetric group s n of degree 5 n 9 are available two observations arise when we look at the results 1 for n 5 the n 1 simplex is up to isomorphism the unique regular n 1 polytope having s n as automorphism group and 2 for n 7 there exists up to isomorphism and duality a unique regular n 2 polytope whose automorphism group is s n we prove that 1 is true for n 4 and 2 is true for n 7 finally we also prove that s n acts regularly on at least one abstract polytope of rank r for every 3 r n 1

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