finite p groups all of whose d generated subgroups are powerful let p be an odd prime we study d powerful p groups i e finite p groups all of whose d generated subgroups are powerful for p 5 we show that any powerful p group g is d g powerful where d g denotes the minimal number of generators of g moreover for a finite p group g we prove that if there is a positive integer 2 m d g such that g is m powerful then all subgroups of g are powerful i e g is modular our results lead to a new characterization of dually strongly balanced finite p groups furthermore minimal non powerful finite p groups are determined completely up to isomorphism for finitely generated pro p groups some analogous results are proved for instance if for a torsion free finitely generated pro p group g there is a positive integer 2 m d g such that all m generated subgroups of g are powerful then all subgroups of g are powerful

an operator without invariant subspaces on a nuclear frechet space let x be a locally convex topological complex vector space throughout this paper the term operator on x will mean a continuous linear transformation mapping x into itself and the term invariant subspace for an operator t on x will mean a closed subspace m of x such that m f o m p x and tm c m it is still an open problem whether or not every operator on an infinite dimensional separable hilbert space has an invariant subspace on the other hand p enflo 8 constructed an operator without invariant subspaces on a nonreflexive banach space in this paper we construct an operator t on an infinite dimensional nuclear frechet space f such that t has no invariant subspaces and t2 has an invariant subspace moreover the space f is a nuclear countably hilbert space in the sense of gelfand and vilenkin 12 that is f is the intersection of a decreasing sequence of hilbert spaces hn n 1 2 such that for every n the imbedding of hn 1 into hn is a nuclear mapping and the topology on f is defined by the sequence of hn norms the elements of f are complex sequences c ci j 0 and the operator t is the associated backward shift that is

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

on the number of conjugacy classes of finite nilpotent groups abstract we establish the first super logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group in particular we obtain that for any constant c there are only finitely many finite p groups of order p m with at most c m conjugacy classes this answers a question of l pyber

hausdorff dimension pro p groups and kac moody algebras every finitely generated profinite group can be given the structure of a metric space and as such it has a well defined hausdorff dimension function in this paper we study hausdorff dimension of closed subgroups of finitely generated pro p groups g we prove that if g is p adic analytic and h c g is a closed subgroup then the hausdorff dimension of h is dimh dimg where the dimensions are of h and g as lie groups letting the spectrum spec g of g denote the set of hausdorff dimensions of closed subgroups of g it follows that the spectrum of p adic analytic groups is finite and consists of rational numbers we then consider some non p adic analytic groups g and study their spectrum in particular we investigate the maximal hausdorff dimension of nonopen subgroups of g and show that it is equal to 1 1 d 1 in the case of g sld fp t where p 2 and to 1 2 if g is the so called nottingham group where p 5 we also determine the spectrum of sl2 fp t p 2 completely showing that it is equal to 0 2 3 1 some of the proofs rely on the description of maximal graded subalgebras of kac moody algebras recently obtained by the authors in joint work with e i zelmanov

groups without faithful transitive permutation representations of small degree abstract a subgroup h of a group g is core free if h contains no non trivial normal subgroup of g or equivalently the transitive permutation representation of g on the cosets of h is faithful we study the obstacles to a group having large core free subgroups we call a subgroup d a dedekind subgroup of g if all subgroups of d are normal in g our main result is the following if a finite group g has no core free subgroups of order greater than k then g has two dedekind subgroups d 1 and d 2 such that every subgroup in g of order greater than f k has non trivial intersection with either d 1 or d 2 where f is a fixed function independent of g examples show that the dedekind subgroups need not have index bounded by a function of k and the result would not be true with one dedekind subgroup instead of two we exhibit various related properties of p groups and infinite locally finite groups without large core free subgroups including the following if g is a locally finite group with no infinite core free subgroup then every infinite subgroup of g contains a non trivial cyclic normal subgroup of g we also exhibit asymptotic bounds for some related problems including the following if a group g has a solvable subgroup of index n then g has a solvable normal subgroup of index at most n c for some absolute constant c if g is a transitive permutation group of degree n with cyclic point stabilizer subgroup then g n 7

uniform groups and lie algebras abstract in this text we study the exp log functors for the categories of uniform groups and uniform lie algebras

on nilpotent multipliers of some verbal products of groups the paper is devoted to finding a homomorphic image for the c nilpotent multiplier of the verbal product of a family of groups with respect to a variety v when v nc or nc v also a structure of the c nilpotent multiplier of a special case of the verbal product the nilpotent product of cyclic groups is given in fact we present an explicit formula for the c nilpotent multiplier of the nth nilpotent product of the group g z n nz nzr1 n nzrt where ri 1 divides ri for all i 1 i t 1 and p r1 1 for any prime p less than or equal to n c for all positive integers n c

on the exponent of a finite group with an automorphism group of order twelve abstract suppose that a frobenius group fh with kernel f of order 4 and complement h of order 3 acts coprimely on a finite group g in such a manner that c g f 1 and c g h has exponent e we prove that the exponent of g is e bounded earlier results of this nature were known only for metacyclic frobenius groups

on the relation between upper central quotients and lower central series of a group let h be a group with a normal subgroup n contained in the upper central subgroup zch in this article we study the influence of the quotient group g h n on the lower central subgroup yc lh in particular for any finite group g we give bounds on the order and exponent of yc l h for g equal to a dihedral group or quaternion group or extra special group we list all possible groups that can arise as yc lh our proofs involve i the baer invariants of g ii the schur multiplier m l g of g relative to a normal subgroup l and iii the nonabelian tensor product of groups some results on the nonabelian tensor product may be of independent interest

on centralizers in locally finite groups abstract the main result of the paper is the following theorem let g be a locally finite group containing a finite p subgroup a such that cg a is finite and a non cyclic subgroup b of order p2 such that cg b has finite exponent for all b b then g is almost locally solvable and has finite exponent

global bv entropy solutions to the gauss codazzi system abstract the gauss codazzi system for the given metric whose gauss curvature is negative is considered in the present paper we provide some suitable conditions on the given metric such that the system admits a global bv entropy solution our main approach is applying the method of seeking global bv entropy solutions to the hyperbolic system of balance laws to the gauss codazzi system the global existence of bv solutions to the gauss codazzi system yields a c 1 1 isometric immersion of two dimensional manifold into r 3

orbital and strongly orbital spaces abstract we say that a countably dimensional topological vector space x is orbital if there is t l x and a vector x x such that x is the linear span of the orbit t n x n 0 1 we say that x is strongly orbital if additionally x can be chosen to be a hypercyclic vector for t of course x can be orbital only if the algebraic dimension of x is finite or infinite countable we characterize orbital and strongly orbital metrizable locally convex spaces we also show that every countably dimensional metrizable locally convex space x does not have the invariant subset property that is there is t l x such that every non zero x x is a hypercyclic vector for t finally assuming the continuum hypothesis we construct a complete strongly orbital locally convex space as a byproduct of our constructions we determine the number of isomorphism classes in the set of dense countably dimensional subspaces of any given separable infinite dimensional frechet space x for instance in x l 2 there are exactly 3 pairwise non isomorphic as topological vector spaces dense countably dimensional subspaces

a characterisation of powerfully embedded normal subgroups of ap group abstract letpbe a prime number andgbe ap group 6 have introduced the notion of a powerfully embedded subgroup ofg our main result gives a characterisation of those powerfully embedded subgroups which are contained in the frattini subgroup ofg whenpis odd whenp 2 the subgroups characterised have a weaker property than powerful embedding this result also yields information on the structure of d maximal 2 groups

hypercyclic weighted shifts an operator t acting on a hilbert space is hypercyclic if for some vector x in the space the orbit tnx n 0 is dense in this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic as a consequence we show within the class of weighted shifts that multi hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic for general hypercyclic operators the corresponding questions were posed by d a herrero and they still remain open using a different technique we prove that i t is hypercycic whenever t is a unilateral backward weighted shift thus answering in more generality a question recently posed by k c chan and j h shapiro

pro finite p adic lie algebras abstract let p be a prime number a finite nilpotent lie ring of characteristic a power of p is called finite p a pro p lie ring is an inverse limit of finite p lie rings pro p lie rings play a role in lie theory similar to that played by pro p groups in group theory every pro p lie ring admits the structure of a lie algebra over the p adic integers furthermore every p adic lie algebra of finite rank as a p adic module has an open pro p subalgebra we make a detailed study of pro p lie rings in terms of various properties including their topology prufer rank subring growth and p adic module structure in particular we prove the equivalence of the following conditions for a finitely generated pro p lie ring l l has finite prufer rank l is isomorphic to a closed subring of gl v for some p adic module v of finite rank and for sufficiently large n the lie f p subalgebra w n e 12 t e 22 gl 2 f p t t n is not an open section of l by reducing to the pro p lie ring case we also prove that all engelian pro finite lie rings are locally nilpotent this is a lie theoretic analogue of zelmanov s theorem which states that every periodic pro p group is locally finite

on pro p groups with potent filtrations abstract in this note we prove that if g is pf group of finite exponent then the exponent of the second homology group h 2 g m divides the exponent of g for every profinite trivial z g module m we introduce the notion of the exponential rank of a pro p group and find a bound for the exponential rank of a pf group

hyperbolic systems of balance laws via vanishing viscosity abstract global weak solutions of a strictly hyperbolic system of balance laws in one space dimension are constructed by the vanishing viscosity method of bianchini and bressan for global existence a suitable dissipativeness assumption has to be made on the production term g under this hypothesis the viscous approximations u that are globally defined solutions to u t a u u x g u u xx satisfy uniform bv bounds exponentially decaying in time furthermore they are stable in l 1 with respect to the initial data finally as 0 u converges in l loc 1 to the admissible weak solution u of the system of balance laws u t f u x g u 0 when a df

bv weak solutions to gauss codazzi system for isometric immersions abstract the isometric immersion problem for surfaces embedded into r 3 is studied via the fluid dynamic framework introduced in chen et al 2010 6 as a system of balance laws of mixed type the techniques developed in the theory of weak solutions of bounded variation in continuum physics are employed to deal with the isometric immersions in the setting of differential geometry the so called bv framework is formed that establishes convergence of approximate solutions of bounded variation to the gauss codazzi system and yields the c 1 1 isometric realization of two dimensional surfaces into r 3 local and global existence results are established for weak solutions of small bounded variation to the gauss codazzi system for negatively curved surfaces that admit equilibrium configurations as an application the case of catenoidal shell of revolution is provided

on profinite groups in which commutators are covered by finitely many subgroups for a family of group words w we show that if g is a profinite group in which all w values are contained in a union of finitely many subgroups with a prescribed property then the verbal subgroup w g has the same property as well in particular we show this in the case where the subgroups are periodic or of finite rank if g contains finitely many subgroups g1 g2 gs of finite exponent e whose union contains all k values in g it is shown that k g has finite e k s bounded exponent if g contains finitely many subgroups g1 g2 gs of finite rank r whose union contains all k values it is shown that k g has finite k r s bounded rank

schur multipliers and power endomorphisms of groups abstract we obtain new bounds for the exponent of the schur multiplier of a given p group we prove that the exponent of the schur multiplier can be bounded by a function depending only on the exponent of a given group as a consequence we show that the exponent of the schur multiplier of any group of exponent four divides eight and that this bound is best possible the notion of the exponential rank of a p group is introduced we show that powerful p groups have exponential rank either zero or one

finite groups with an automorphism of prime order whose centralizer has small rank abstract let be an automorphism of prime order p of a finite group g and let c g be its fixed point subgroup when is regular that is c g 1 the group g is nilpotent by thompson s theorem the almost regular results of fong and hartley meixner pettet were giving the existence of a nilpotent subgroup of index bounded in terms of p and c g we prove the rank analogues of these results when almost regular in the hypothesis is interpreted as a restriction on the rank r of c g and the conclusion is sought as nilpotency modulo certain bits of bounded rank the classification is used to prove almost solubility in the coprime case the rank of g s g is bounded in terms of r and p for soluble groups the hall higman type theorems are combined with the theory of powerful q groups to obtain almost nilpotency even without the coprimeness condition there are characteristic subgroups r n g such that n r is nilpotent and the ranks of r and g n are bounded in terms of r and p examples show that our results are in a sense best possible

profinite groups and the fixed points of coprime automorphisms abstract the main result of the paper is the following theorem let q be a prime and a an elementary abelian group of order q 3 suppose that a acts coprimely on a profinite group g and assume that c g a is locally nilpotent for each a a then the group g is locally nilpotent

the unitary cover of a finite group and the exponent of the schur multiplier abstract for a finite group we introduce a particular central extension the unitary cover having minimal exponent among those satisfying the projective lifting property we obtain new bounds for the exponent of the schur multiplier relating to subnormal series and we discover new families for which the bound is the exponent of the group finally we show that unitary covers are controlled by the zelmanov solution of the restricted burnside problem for 2 generator groups

hereditarily hypercyclic operators abstract we show that a continuous linear operator t on a frechet space satisfies the so called hypercyclicity criterion if and only if it is hereditarily hypercyclic and if and only if the direct sum t t is hypercyclic in particular hypercyclic operators with either a dense generalized kernel or a dense set of periodic points i e chaotic in the sense of r l devaney 1989 an introduction to chaotic dynamical systems addison wesley reading ma must satisfy the criterion finally we provide a characterization of those weighted shifts t that are hereditarily hyper cyclic with respect to a given sequence n k of positive integers as well as conditions under which t and t n k k 1 share the same set of hypercyclic vectors

on quaternion free 2 groups abstract two theorems are proved the first of them showing that a modular quaternion free finite 2 group has a characteristic abelian subgroup with metacyclic factor the second classifying nonmodular finite quaternion free 2 groups

hypercyclic and cyclic vectors abstract let x denote an arbitrary separable banach space over the field of complex numbers and b x the banach algebra of all bounded linear operators on x we prove the following results 1 an element of the space x is hypercyclic supercyclic for all positive powers tn of an operator t in b x if it is hypercyclic supercyclic for t 2 under some condition on the spectrum of the adjoint of a cyclic operator the set of all cyclic vectors of the operator is dense this result extends to any cyclic commutative subset of b x 3 under a mild condition on the spectrum of a cyclic operator t the set of all separating vectors for the commutant t of t is dense this also extends to any cyclic commutative subset of b x 4 a slightly stronger version of a theorem of k f clancey and d d rogers on cyclic vectors finally we define and discuss hereditarily hypercyclic operators

powerful p groups have non inner automorphisms of order p and some cohomology in this paper we study the longstanding conjecture of whether there exists a non inner automorphism of order p for a finite non abelian p group we prove that if g is a finite non abelian p group such that g z g is powerful then g has a non inner automorphism of order p leaving either g or 1 z g elementwise fixed we also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for odd p by showing that the tate cohomology hn g n z n 0 for all n 0 where g is a finite p group p is odd g z g is p central i e elements of order p are central and n g with g n non cyclic

some computations of non abelian tensor products of groups on considere le produit tensoriel generalise g h des groupes g h on presente le calcul de g g

a bound for the exponent of the schur multiplier of a finite p group let p be a prime g a finite p group r the rank of g and e the exponent of g in this short note we will prove that the exponent of the schur multiplier of g divides pe r log2 rlog2p 1p 1

a non abelian free pro pgroup is not linear over a local field abstract in this paper we show that a non abelian free pro pgroup cannot be obtained as a closed subgroup ofgln f wherefis a non archimedean local field andnis arbitrary using a theorem of e i zel manov we deduce some group theoretic properties of linear pro pgroups over a local field our main tool is a recent theorem by pink characterizing compact subgroups ofgln f

existence of hypercyclic operators on topological vector spaces the present paper introduces a very simple but very useful notion of the so called quasi extension ofl1 operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators in particular it answers in the affirmative a question of s rolewicz posed in 1969 whether or not every infinite dimensional separable banach space admits a continuous hypercyclic operator

on p saturable groups a pro p group g is a pf group if it has central series of closed subgroups ni i n with trivial intersection satisfying n1 g and ni g p 1 g ni 1p in this paper we prove that a finitely generated pro p group g is a p saturable group in the sense of lazard if and only if it is a torsion free pf group using this characterization we study certain families of subgroups of p saturable groups for example we prove that any normal subgroup of a p saturable group contained in the frattini is again p saturable

isometric immersions and compensated compactness a fundamental problem in differential geometry is to characterize intrinsic metrics on a two dimensional riemannian manifold mathcal m 2 which can be realized as isometric immersions into mathbb r 3 this problem can be formulated as initial and or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic hyperbolic type whose mathematical theory is largely incomplete in this paper we develop a general approach which combines a fluid dynamic formulation of balance laws for the gauss codazzi system with a compensated compactness framework to deal with the initial and or boundary value problems for isometric immersions in mathbb r 3 the compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the gauss codazzi system for approximate solutions which yields the isometric realization of two dimensional surfaces in mathbb r 3

powerful 2 engel groups ii we conclude our classification of powerful 2 engel groups of class three that are minimal in the sense that every proper powerful section is nilpotent of class at most two in the predecessor to this paper we obtained three families of minimal groups here we get a fourth family of minimal examples that is described in terms of irreducible polynomials over the field of three elements we also get one isolated minimal example of rank 5 and exponent 27 the last one has a related algebraic structure that we call a symplectic alternating algebra to each symplectic alternating algebra over the field of three elements there corresponds a unique 2 engel group of exponent 27

hypercyclic operators on non normable frechet spaces abstract every infinite dimensional separable non normable frechet space admits a continuous hypercyclic operator a large class of separable countable inductive limits of banach spaces with the same property is given but an example of a separable complete inductive limit of banach spaces which admits no hypercyclic operator is provided it is also proved that no compact operator on a locally convex space is hypercyclic

on elements of order p in powerful p groups abstract we investigate the set 1 p of elements of order at most p in a powerful p group p and prove that 1 p p p p as a corollary we obtain a necessary and sufficient condition for 1 p to be of exponent p we give an example to show that for p 2 there is a powerful 2 group such that 1 p is not of exponent 2

characterization of p adic analytic groups in terms of wreath products abstract we show that a finitely generated pro p group is p adic analytic i e can be given the structure of a lie group over q p if and only if it does not involve arbitrarily large wreath products of the form c p wr c p n this result whose proof applies zelmanov s recent solution to the restricted burnside problem is in fact equivalent to zelmanov s theorem

bounding the exponent of a finite group with automorphisms a 4 where a s a 1 this follows immediately from the fact that for every prime p dividing g there is an a invariant sylow p subgroup p of g for a which even a stronger decomposition holds p s c a see for ag a p w x example 6 theorems 6 2 2 and 6 2 4 suppose that in this situation n is the least common multiple of the exponents of the centralizers c a g a g a we show that then the exponent of g is bounded by a number that depends only on n and on the group a

fixed points of coprime operator groups abstract let m be a positive integer and a an elementary abelian group of order q r with r 2 acting on a finite q group g we show that if for some integer d such that 2 d r 1 the dth derived group of c g a has exponent dividing m for any a a then g d has m q r bounded exponent and if r 1 c g a has exponent dividing m for any a a then r 1 g has m q r bounded exponent

a classification of finite 2 groups with exactly three involutions we finish here a classification of finite 2 groups with exactly three involutions we clear up this remaining very difficult situation and determine completely the structure of g in terms of two generators and relations

polynomial identities in graded group rings restricted lie algebras and p adic analytic groups let g be any finitely generated group and let k be a field of characteristic p 0 it is shown that the graded group ring gr kg satisfies a nontrivial polynomial identity if and only if the pro p completion of g is p adic analytic i e can be given the structure of a lie group over the p adic field q p the proof applies theorems of lazard quillen and passman as well as results on engel lie algebras and on dimension subgroups in positive characteristic

autocommutator subgroups of finite groups abstract we present some new results about the absolute center and autocommutator subgroup of a group concepts which were introduced in an earlier paper

dimension subgroups nilpotency indices and the number of generators of ideals in p group algebras abstract various problems in modular p group algebras are solved through extensive study of dimension subgroups as an application it is shown that if g is an infinite res p group p 2 and k is a field of characteristic p then the least upper bound on the numbers of generators of ideals in kg equals the minimal index of a cyclic subgroup of g

locally finite groups with all subgroups normal by finite rank abstract a group is said to have finite special rank sif all of its finitely generated subgroups can be generated byselements letgbe a locally finite group and suppose thath hghas finite rank for all subgroupshofg wherehgdenotes the normal core ofhing we prove that thenghas an abelian normal subgroup whose quotient is of finite rank theorem 5 if in addition there is a finite numberrbounding all of the ranks ofh hg thenghas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly theorem 4 these results are based on analogous theorems on locally finitep groups in which case the groupgis also abelian by finite theorems 2 and 3

universal elements for non linear operators and their applications abstract we prove that under certain topological conditions on the set of universal elements of a continuous map t acting on a topological space x that the direct sum t m g is universal where m g is multiplication by a generating element of a compact topological group we use this result to characterize r supercyclic operators and to show that whenever t is a supercyclic operator and z 1 z n are pairwise different non zero complex numbers then the operator z 1 t z n t is cyclic the latter answers affirmatively a question of bayart and matheron

extension of a schur theorem to groups with a central factor with a bounded section rank abstract a well known result reported by schur states that the derived subgroup of a group is finite provided its central factor is finite here we show that if the p section rank of the central factor of a locally generalized radical group is bounded then so is the p section rank of its derived subgroup we also give an explicit expression for this bound

some baer invariants of free nilpotent groups abstract we present an explicit structure for the baer invariant of a free nth nilpotent group the nth nilpotent product of infinite cyclic groups z n z n n z with respect to the variety v with the set of words v c 1 1 c 2 1 for all c 1 c 2 and 2 c 2 c 1 2 n 2 also an explicit formula for the polynilpotent multiplier of a free nth nilpotent group is given for any class row c 1 c 2 c t where c 1 n

on the growth of subalgebras in lie p algebras abstract let l be a finitely generated lie p algebra over a finite field f then the number a n l of p subalgebras of finite codimension n in l is finite we say that l has psg polynomial p subalgebras growth if the growth of a n l is bounded above by some polynomial in f n we show that if l has psg then the lower central series of l stabilises after a finite number of steps on the other hand if l is nilpotent then l has psg we deduce the following group theoretic result let g be a group and let g p denote a pro p completion of g then the associated lie p algebra l p g of g has psg if and only if g p is a p adic analytic lie group

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

on the structure of normal subgroups of potent p groups abstract let g be a finite p group satisfying g g g4 for p 2 and p 1 g gp for p 2 the main goal of this paper is to show that any normal subgroup n of g lying in g2 is power abelian that is the following holds 1 npk gpk g n 2 k n g n o g p k and 3 n p k n k n

beauville structures in finite p groups abstract we study the existence of unmixed beauville structures in finite p groups where p is a prime first of all we extend catanese s characterisation of abelian beauville groups to finite p groups satisfying certain conditions which are much weaker than commutativity this result applies to all known families of p groups with a good behaviour with respect to powers regular p groups powerful p groups and more generally potent p groups and generalised p central p groups in particular our characterisation holds for all p groups of order at most p p which allows us to determine the exact number of beauville groups of order p 5 for p 5 and of order p 6 for p 7 on the other hand we determine which quotients of the nottingham group over f p are beauville groups for an odd prime p as a consequence we give the first explicit infinite family of beauville 3 groups and we show that there are beauville 3 groups of order 3 n for every n 5

groups satisfying semigroup laws and nilpotent by burnside varieties abstract we investigate the structure of groups satisfying apositive law that is an identity of the formu v whereuandvare positive words the main question here is whether all such groups are nilpotent by finite exponent we answer this question affirmatively for a large class c of groups including soluble and residually finite groups showing that moreover the nilpotency class and the finite exponent in question are bounded solely in terms of the length of the positive law it follows in particular that if a variety of groups is locally nilpotent by finite then it must in fact be contained in the product of a nilpotent variety by a locally finite variety of finite exponent we deduce various other corollaries for instance that a torsion free residually finite n engel group is nilpotent of class bounded in terms ofn we also consider incidentally a question of bergman as to whether a positive law holding in a generating subsemigroup of a group must in fact be a law in the whole group showing that it has an affirmative answer for soluble groups

on the existence of c 1 1 isometric immersions of several classes of negatively curved surfaces into mathbb r 3 we prove the existence of c 1 1 isometric immersions of several classes of negatively curved riemannian surfaces mathcal m g into the 3 dimensional euclidean space mathbb r 3 including the standard hyperbolic plane the generalised helicoid type metrics and the generalised enneper metrics our proof is based on the theory of compensated compactness and the method of invariant regions in hyperbolic conservation laws together with several new observations on relevant geometric quantities gauss curvature metric components etc for negatively curved surfaces

asymptotic results for primitive permutation groups abstract we prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostnc n where n denotes the maximal exponent occurring in the prime factorization ofn this result is applied to investigating maximal subgroup growth of infinite groups we then proceed by showing that if the point stabilizerg of a primitive groupgof degreendoes not have the alternating group alt d as a section then the order ofgis bounded by a polynomial inn this result extends a well known theorem of babai cameron and palfy it is used to prove for example that ifhis a subgroup of indexnin a groupg andhis a product ofbcyclic groups then g hg ncwherecdepends onb

isometric immersions of surfaces with two classes of metrics and negative gauss curvature the isometric immersion of two dimensional riemannian manifolds or surfaces with negative gauss curvature into the three dimensional euclidean space is studied in this paper the global weak solutions to the gauss codazzi equations with large data in l infty are obtained through the vanishing viscosity method and the compensated compactness framework the l infty uniform estimate and h 1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type the global weak solutions in l infty to the gauss codazzi equations yield the c1 1 isometric immersions of surfaces with the given metrics