finite groups with biprimary hall subgroups abstract let g be a finite group and let r be a prime divisor of the order of g we prove that if r 5 and g has the e r t property for all t g r then g is r solvable a group g is said to have the e property if g possesses a hall subgroup

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

sylow numbers and nilpotent hall subgroups let be a set of primes and g a finite group we characterize the existence of a nilpotent hall subgroup of g in terms of the number of sylow subgroups for the primes in

grid structures and undecidable constraint theories abstract we prove three new undecidability results for computational mechanisms over finite trees there is a linear ultra shallow noetherian and strongly confluent rewrite system r such that the fragment of the first order theory of one step rewriting by r is undecidable the emptiness problem for tree automata with equality tests between cousins is undecidable and the fragment of the first order theory of set constraints with the union operator is undecidable the common feature of these three computational mechanisms is that they allow us to describe the set of first order terms that represent grids we extend our representation of grids by terms to a representation of linear two dimensional patterns by linear terms which allows us to transfer classical techniques on the grid to terms and thus to obtain our undecidability results

confirmation for wielandt s conjecture abstract let be a set of primes by h wielandt s definition sylow theorem holds for a finite group g if all maximal subgroups of g are conjugate in the paper the following statement is proven assume that is a union of disjoint subsets and and a finite group g possesses a hall subgroup which is a direct product of a subgroup and a subgroup furthermore assume that both the sylow theorem and theorem hold for g then the sylow theorem holds for g this result confirms a conjecture posed by h wielandt in 1959

termination of narrowing revisited this paper describes several classes of term rewriting systems trs s where narrowing has a finite search space and is still strongly complete as a mechanism for solving reachability goals these classes do not assume confluence of the trs we also ascertain purely syntactic criteria that suffice to ensure the termination of narrowing and include several subclasses of popular trs s such as right linear trs s almost orthogonal trs s topmost trs s and left flat trs s our results improve and or generalize previous criteria in the literature regarding narrowing termination

set constraints in some equational theories set constraints are relations between sets of ground terms over a given alphabet they give a natural formalism for many problems in program analysis type inference order sorted unification and constraint logic programming in this paper we start studies of set constraints in the environment given by equational specifications we show that in the case of associativity i e in free monoids as well as in the case of associativity and commutativity i e in commutative monoids the problem of consistency of systems of set constraints is undecidable in linear nonerasing shallow theories the consistency of systems of positive set constraints is nexptime complete and in linear shallow theories the problem for positive and negative set constraints is decidable

decidability and complexity analysis by basic paramodulation it is shown that for sets of horn clauses saturated underbasic paramodulationthe word and unifiability problems are in np and the number of minimal unifiers is simply exponential i for horn sets saturated w r t a special ordering under the more restrictive inference rule ofbasic superposition the word and unifiability problems are still decidable and unification is finitary ii these two results are applied to the following languages forshallowpresentations equations with variables at depth at most one we show that the closure under paramodulation can be computed in polynomial time applying result i it follows that shallow unifiability is in np which is optimal since unifiability in ground theories is already np hard the shallow word problem is even shown to be polynomial generalizing shallow theories to the horn case we obtain two versions of a language we callcatalog a natural extension of datalog to include functions and equality the closure under paramodulation is finite for catalog sets hence i still applies for catalog sets s the decidability of the full first order theory of t f sis shown as well finally we definestandard theories which include and significantly extendshallowtheories standard presentations can be finitely closed under superposition and result ii applies thus obtaining a new fundamental class with decidable word and unifiability problems and where unification is finitary

syntacticness cycle syntacticness and shallow theories abstract solving equations in the free algebra t f x i e unification uses the two rules s t s t decomposition and s x x occur check these two rules are not correct in quotients of t f x by a finitely generated congruence e following c kirchner we first define classes of equational theories called syntactic and cycle syntactic respectively for which it is possible to derive some rules replacing the two above then we show that these abstract classes are relevant all shallow theories i e theories which can be generated by equations in which variables occur at depth at most one are both syntactic and cycle syntactic moreover the new set of unification rules is terminating which proves that unification is decidable and finitary in shallow theories we give still further extensions if the set of equivalence classes is infinite a problem which turns out to be decidable in shallow theories then shallow theories fulfill colmerauer s independence of disequations principle a conjunction of n disequations is solvable iff each disequation alone is solvable this allows us to derive quantifier elimination rules it turns out that these rules do terminate for shallow theories hence the first order theory of the quotient algebra t f e is decidable when f is finite and e is shallow this extends mal cev results on the classes of permutative locally free algebras that are completely axiomatizable

constructive sylow theorems for the classical groups abstract let r be a prime we provide efficient algorithms to construct and solve the conjugacy problem for the sylow r subgroups of the classical groups over finite fields in their natural representations we also provide algorithms to construct the normaliser of a sylow r subgroup of gl n f q sp 2 m f q go 2 m 1 f q go 2 m f q go 2 m f q or gu n f q 2 the algorithms described here are implemented in the magma computer algebra system

holder regularity for certain strongly coupled parabolic systems abstract we present a shorter proof to show holder continuity of bounded solutions to a general class of quasilinear parabolic equations the proof will be extended to obtain regularity results for bounded solutions to certain strongly coupled or cross diffusion quasilinear parabolic systems

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

extensions of unipotent characters and the inductive mckay condition abstract we prove that unipotent characters of groups of lie type extend to their inertia groups in the full automorphism group we then apply this in order to show that certain groups of lie type are good for the prime p 3 or p 2 this deals with all the exceptions not covered by our general approach for verifying the mckay conjecture for all groups of lie type in non defining characteristic

block separations and inclusions abstract we investigate the separation of characters by blocks at different primes and the inclusions of q blocks in p blocks viewed as sets of characters and use these notions to prove results on the structure of the corresponding groups in particular we provide a new criterion for the nilpotence of a finite group g based on the separation by principal blocks and we show that a condition on block unions has strong structural consequences

small intersections of principal blocks abstract in this note it is shown that a finite group g is solvable if for each odd prime divisor p of g irr b 0 g 2 irr b 0 g p 2 where irr b 0 g p is the set of complex irreducible characters of the principal p block b 0 g p of g also the structure of such groups is investigated examples show that the bound 2 is best possible

global attractors and steady state solutions for a class of reaction diffusion systems abstract we show that weak l p dissipativity implies strong l dissipativity and therefore implies the existence of global attractors for a general class of reaction diffusion systems this generalizes the results of alikakos and rothe the results on positive steady states especially for systems of three equations in our earlier work j differential equations 130 1996 59 91 are improved

on the number of classes of conjugate hall subgroups in finite simple groups abstract in this paper we find the number of classes of conjugate hall subgroups in all finite almost simple groups we also complete the classification of hall subgroups in finite simple groups and correct some mistakes from our previous paper