a question of r maier concerning formations the formation f is said to be saturated if the group g belongs to f whenever the frattini factor group grf g is in f let p be the set of all prime numbers a formation function is a function f defined on p such that f p is a possibly empty formation a formation f is said to be a local formation if there exists a formation function f such that f s g g g if hrk is a chief factor of g and p divides hrk then grc hrk g f p g is the class of all finite g groups if f is a local formation defined by a formation function f then we denote f s lf f and f is a local definition of f among all possible local definitions of a local formation f there exists exactly one denoted by f such that f is integrated i e f p f for all p g p and full s f p s f p for all p g p here s denotes the class of all p groups p p this f is the canonical local definition of f the celebrated w x gaschutz lubeseder schmid theorem 2 iv theorem 4 6 states that a formation is saturated if and only if it is local let f be a formation it is known that each group g has a smallest normal subgroup with quotient in f it is called the f residual of g and is denoted by g on the other hand it is well known that formations are

the equivalence of some conjectures of dade and robinson we demonstrate that conjecture 4 1 of g r robinson proc london math soc 3 72 1996 312 330 and dade s projective conjecture are equivalent in a way which is compatible with the p local rank further we consider refinements of these conjectures similar to those of isaacs navarro and uno show their equivalence and demonstrate that in order to verify them it suffices to consider only those groups with no non central normal p subgroup

on finite groups generated by strongly cosubnormal subgroups two subgroups a and b of a group g are cosubnormal if a and b are subnormal in their join a band are strongly cosubnormal if every subgroup of a is cosubnormal with every subgroup of b we find necessary and sufficient conditions for a and b to be strongly cosubnormal ina b and if z is the hypercentre of g a b we show that a and b are strongly cosubnormal if and only if g z is the direct product of az z and bz z we also show that projectors and residu als for certain formations can easily be constructed in such a group two subgroups a and b of a group g are n connected if every cyclic subgroup of a is cosubnormal with every cyclic subgroup of b n denotes the class of nilpotent groups though the concepts of strong cosubnormality and n connectedness are clearly closely related we give an example to show that they are not equivalent we note however that if g is the product of the n connected subgroups a and b t hena and b are strongly cosubnormal

broue s abelian defect group conjecture holds for the harada norton sporadic simple group hn abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding block b of the normaliser n g p of p in g are derived equivalent rickard equivalent this conjecture is called broue s abelian defect group conjecture we prove in this paper that broue s abelian defect group conjecture is true for a non principal 3 block a with an elementary abelian defect group p of order 9 of the harada norton simple group hn it then turns out that broue s abelian defect group conjecture holds for all primes p and for all p blocks of the harada norton simple group hn

character correspondences in blocks with normal defect groups abstract in this paper we give an extension of the glauberman correspondence to certain characters of blocks with normal defect groups

totally permutable products of certain classes of finite groups abstract subgroups a and b of a finite group are said to be totally permutable if every subgroup of a permutes with every subgroup of b the behaviour of finite pairwise totally permutable products are studied with respect to certain classes of groups including the class of groups where all the subnormal subgroups permute with all the maximal subgroups the so called sm groups and also the class of groups where all the subnormal subgroups are permutable the so called pt groups

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

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

a reduction theorem for the mckay conjecture the mckay conjecture asserts that for every finite group g and every prime p the number of irreducible characters of g having p degree is equal to the number of such characters of the normalizer of a sylow p subgroup of g although this has been confirmed for large numbers of groups including for example all solvable groups and all symmetric groups no general proof has yet been found in this paper we reduce the mckay conjecture to a question about simple groups we give a list of conditions that we hope all simple groups will satisfy and we show that the mckay conjecture will hold for a finite group g if every simple group involved in g satisfies these conditions also we establish that our conditions are satisfied for the simple groups psl2 q for all prime powers q 4 and for the suzuki groups sz q and ree groups r q where q 2 e or q 3 e respectively and e 1 is odd since our conditions are also satisfied by the sporadic simple group j 1 it follows that the mckay conjecture holds for all primes p for every finite group having an abelian sylow 2 subgroup

refinements of dade s projective conjecture for p solvable groups dade s projective conjecture is known to be true for finite p solvable groups thanks to work of g r robinson but remains open in general work of isaacs and navarro suggested to uno and boltje refinements of this conjecture these refinements were studied for finite p solvable groups by glesser in the present paper inspired by earlier work of turull we propose further refinements of the conjecture that take into account the schur indices and the elements of the brauer group we prove that all these refinements of dade s projective conjecture hold for all finite p solvable groups in particular we obtain that the version of dade s projective conjecture which involves character degree residues modulo p fields of definition and schur indices as well as the full strength of boltje s conjecture both hold for all finite p solvable groups the proof develops a clifford theory for normalizers of chains of p subgroups which allows one to reduce the calculation of the relevant sums to simpler 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 two variable rational interpolation abstract the goal of this contribution is to investigate interpolation of two variable rational functions the tool is the two variable loewner matrix which is an extension of its single variable counterpart the main property of the loewner matrix is that its rank encodes the information concerning minimal complexity interpolants both polynomial and generalized state space descriptor realizations of interpolants are presented examples illustrate how two variable rational functions can be recovered from appropriate measurements

strengthening the mckay conjecture to include local fields and local schur indices abstract a central problem in the representation theory of finite groups is given a prime p the study of relationships between the representations of g and those of p local subgroups of g the simplest of these is perhaps the mckay conjecture which concerns the number of irreducible characters of degree prime to p even though this conjecture remains open in an effort to understand it a number of strengthenings of it have been proposed recently isaacs and navarro and later navarro have proposed interesting stronger conjectures indicating that congruences modulo p of the degree of the characters and the invariance under certain type of fixed galois automorphism could be taken into account as well we propose an even further strengthening of mckay s conjecture which takes into account in addition the p local field of definition of the irreducible characters as well as their p local schur indices

on totally permutable products of finite groups this paper has been published in journal of algebra 293 1 269 278 2005 copyright 2005 by elsevier http dx doi org 10 1016 j jalgebra 2005 01 033

geometric methods in rational interpolation theory abstract this paper studies the minimal rational interpolation problem using algebrogeometric methods specifically we deal in a unified framework with the following problems scalar and matrix interpolation and left and right tangential interpolation while the results we prove are known our methods of proof are quite different in addition our proof shows that these problems can be solved by an application of the grobner basis algorithm a popular computational tool in computer algebra

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

some refinements of dade s projective conjecture abstract in this paper a further refinement of dade s projective conjecture due to boltje is presented this new statement includes ideas first published by isaacs and navarro as well as the recent contractibility version of alperin s conjecture introduced by boltje leaning heavily on the work of robinson weaker forms of the conjecture are proved in the case of p solvable groups

on clifford theory with galois action abstract let g be a finite group n a normal subgroup of g and irr n let f be a subfield of the complex numbers and assume that the galois orbit of over f is invariant in g we show that there is another triple g 1 n 1 1 of the same form such that the character theories of g over and of g 1 over 1 are essentially the same over the field f and such that the following holds g 1 has a cyclic normal subgroup c contained in n 1 such that 1 n 1 for some linear character of c and such that n 1 c is isomorphic to the abelian galois group of the field extension f f 1 more precisely having the same character theory means that both triples yield the same element of the brauer clifford group brcliff g f defined by a turull

on a generalisation of totally permutable products of finite groups abstract a group g a b is a weakly totally permutable product of subgroups a and b if for every subgroup u of a such that u a b or a b u permutes with every subgroup of b and if for every subgroup v of b such that v a b or a b v permutes with every subgroup of a results on totally permutable products are extended to weakly totally permutable products in particular it is shown that for a weakly totally permutable product if the factors are in f then the product is also in f where f is a formation containing u the class of finite supersoluble groups it is also shown that f residuals and f projectors behave nicely in weakly totally permutable products when f is a saturated formation containing u

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

height zero characters and normal subgroups in p solvable groups abstract fix a prime number p and let n be a normal subgroup of a finite p solvable group g let b be a p block of n and suppose b is a p block of g covering b let d be a defect group for the fong reynolds correspondent of b with respect to b and let b be the unique p block of n n g d having defect group d and inducing b suppose further that irr b and let irr 0 b be the set of irreducible characters in b of height zero that lie over we show that the number of characters in irr 0 b is equal to the number of characters in t irr 0 b t where t runs through the inertial group t of b in g this result generalizes a theorem of t okuyama and m wajima which confirms the alperin mckay conjecture for p solvable groups

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

the strengthened alperin weight conjecture for p solvable groups abstract the alperin weight conjecture is a well known conjecture and it is central to the modern representation theory of finite groups it is known to be true for many types of finite groups but it remains open in particular the conjecture is known to be true for finite p solvable groups in this paper we prove that a very strong form of the conjecture holds for all finite p solvable groups it follows that the strengthening of the alperin weight conjecture that includes galois automorphisms over the p adic numbers holds for all finite p solvable groups this strengthening of the alperin weight conjecture was suggested in an earlier work of navarro on the mckay conjecture the present paper provides the first direct evidence beyond the calculation of individual cases and the case of groups of odd order for this strengthening

block based thiele like blending rational interpolation thiele type continued fraction interpolation may be the favoured nonlinear interpolation in the sense that it is constructed by means of the inverse differences which can be calculated recursively and produce useful intermediate results however thiele s interpolation is in fact a point based interpolation by which we mean that a new interpolating continued fraction with one more degree of its numerator or denominator polynomial is obtained by adding a tail to the current one or in other words this can be reshaped simply by adding a new support point to the current set of support points once at a time in this paper we introduce so called block based inverse differences to extend the point based thiele type interpolation to the block based thiele like blending rational interpolation the construction process may be outlined as follows first of all divide the original set of support points into some subsets blocks then construct each block by using whatever interpolation means linear or rational and finally assemble these blocks by thiele s method to shape the whole interpolation scheme clearly many flexible rational interpolation schemes can be obtained in this case including the classical thiele s continued fraction interpolation as its special case as an extension a bivariate analogy is also discussed and numerical examples are given to show the effectiveness of our method

galois group action and jordan decomposition of characters of finite reductive groups with connected center abstract let g be a connected reductive group with connected center defined over f q with frobenius morphism f given an irreducible complex character of g f with its jordan decomposition and a galois automorphism gal q q we give the jordan decomposition of the image of under the action of on its character values

fitting classes and products of totally permutable groups the second and third authors have been supported by proyecto pb 97 0674 c02 02 of dgesic ministerio de educacion y cultura spain

character correspondences in solvable groups let g be a finite solvable group p be some prime let p be a sylow p subgroup of g and let n be its normalizer in g it is known that the number of irreducible characters of g of degree prime to p equals the number of irreducible characters of n of degree prime to p let f be any field of characteristic zero such that if g n is even then f contains qp q 2 the intersection of the field of p adic numbers with the field of rational numbers extended by all roots of unity whose order is a power to two in this paper we show that there exists a bijection from the set of all irreducible characters of g of degree prime to p to the set of all the irreducible characters of degree prime to p of n such that it preserves the degrees modulo p all the field of values over f and the schur index over every field containing f

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

uno s invariant conjecture for chevalley groups g2 q in nondefining characteristics abstract uno s invariant conjecture for the chevalley groups g 2 q has been verified when the characteristic of the modular representation is distinct from the defining characteristic of the groups thus dade s reductive conjecture and the isaacs navarro conjecture both hold for g 2 q in non defining characteristics

above the glauberman correspondence let s be a finite solvable group and suppose s acts on the finite group n and they have coprime orders then the celebrated glauberman correspondence provides a natural bijection from the set irrs n of irreducible characters of n which are invariant under the action of s to the set irr cn s of all irreducible characters of the centralizer of s in n suppose further that the semidirect product sn is a normal subgroup of a finite group g let irrs n and let irr cn s be its glauberman correspondent we prove that there is a bijection with good compatibility properties from the set irr g of the irreducible characters of g above to irr ng s such that in the case when s is a p group for some prime p it preserves fields of values and schur indices over qp the field of p adic numbers using this result we also prove a strengthening of the mckay conjecture for all p solvable groups

on finite products of groups and supersolubility abstract two subgroups x and y of a group g are said to be conditionally permutable in g if x permutes with y g for some element g g i e x y g is a subgroup of g using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered also the behaviour of the supersoluble residual in products of finite groups is studied

new refinements of the mckay conjecture for arbitrary finite groups let g be an arbitrary finite group and fix a prime number p the mckay conjecture asserts that g and the normalizer in g of a sylow p subgroup have equal numbers of irreducible characters with degrees not divisible by p the alperin mckay conjecture is version of this as applied to individual brauer p blocks of g we offer evidence that perhaps much stronger forms of both of these conjectures are true

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

2 brauer correspondent blocks with one simple module abstract one of the main problems in representation theory is to understand the exact relationship between brauer corresponding blocks of finite groups the case where the local correspondent has a unique simple module seems key we study this situation for 2 blocks

fields of values of odd degree irreducible characters abstract in this paper we clarify the quadratic irrationalities that can be admitted by an odd degree complex irreducible character of an arbitrary finite group write q to denote the field generated over the rational numbers by the values of and let d 1 be a square free integer we prove that if q q d then d 1 mod 4 and if q q d then d 3 mod 4 this follows from the main result of this paper either i q or q q exp 2 i m for some odd integer m 1

conditional permutability of subgroups and certain classes of groups abstract two subgroups a and b of a finite group g are said to be tcc permutable if x permutes with y g for some g x y for all x a and all y b some aspects about the normal structure of a product of two tcc permutable subgroups are analyzed the obtained results allow to study the behaviour of such products in relation with certain classes of groups namely the class of t groups and some generalizations

clifford theory and galois theory i abstract we combine galois theory with crossed products by letting grading groups act on ground fields we show that clifford theory for suitable normal subgroups preserves such actions

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

groups whose real irreducible characters have degrees coprime to p abstract in this paper we study groups for which every real irreducible character has degree not divisible by some given odd prime p

newton pade approximations for multivariate functions abstract we modify cuyt and verdonk s approach to multivariate newton pade approximations explicit formulas are given for coefficients which can be computed once for given system of nodes and the linear system of equations to find multivariate newton pade approximants can be written using simple formulas with those coefficients and the newton series of interpolated function