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

on p stability in groups and fusion systems the aim of this paper is to generalise the notion of p stability p is an odd prime in finite group theory to fusion systems we first compare the different definitions of p stability for groups and examine properties of p stability concerning subgroups and factor groups motivated by glauberman s theorem we study the question of how qd p is involved in finite simple groups we show that with a single exception a simple group involving qd p has a subgroup isomorphic to either qd p or a central extension of qd p by a cyclic group of order p then we define p stability for fusion systems and characterise some of its properties we prove a fusion theoretic version of thompson s maximal subgroup theorem we introduce the notion of section p stability both for groups and fusion systems and prove a version of glauberman s theorem to fusion systems we also examine relationship between solubility and p stability for fusion systems and determine the simple groups whose fusion systems are qd p free

a poisson limit theorem for a strongly ergodic non homogeneous markov chain a strongly ergodic non homogeneous markov chain is considered in the paper as an analog of the poisson limit theorem for a homogeneous markov chain recurring to small cylindrical sets a poisson limit theorem is given for the non homogeneous markov chain meanwhile some interesting results about approximation independence and probabilities of small cylindrical sets are given

on the automorphisms of 2 v k 1 designs thirty years ago a six person team classified the pairs d g where d is a 2 v k 1 design and g is a flag transitive automorphism group of d with the exception of those in which g is a one dimensional affine group since then the effort has been to classify those d designs which are block transitive but not flag transitive this paper contributes to the program for determining the pairs 2 v k 1 in which d g has a block transitive group g of automorphisms it is clear that if one wishes to study the structure of a finite group acting on a d design then describing the socle is an important first step here we prove that if g is a block transitive group of automorphisms of 2 v k 1 which has d as its socle then t is also transitive on the blocks of t 2 f 4 q

on the cohomology of the finite chevalley groups abstract let g be a group of lie type let k be an algebraically closed field and let v be a faithful irreducible kg module the aim of this paper is to provide bounds for dim h 1 g v as a function of dim v

entry times distribution for dynamical balls on metric spaces we show that the entry and return times for dynamical balls bowen balls is exponential for systems that have an alpha mixing invariant measure with certain regularities we also show that systems modeled by young s tower has exponential entry time distribution for dynamical balls we also apply the results to conformal maps and expanding maps on the interval

criteria for solvable radical membership via p elements abstract guralnick kunyavskii plotkin and shalev have shown that the solvable radical of a finite group g can be characterized as the set of all x g such that x y is solvable for all y g we prove two generalizations of this result firstly it is enough to check the solvability of x y for every p element y g for every odd prime p secondly if x has odd order then it is enough to check the solvability of x y for every 2 element y g

reduced fusion systems over p groups with abelian subgroup of index p ii abstract for an odd prime p we look at simple fusion systems over a finite nonabelian p group s which has an abelian subgroup a of index p when s has more than one such subgroup we reduce this to a case already studied by ruiz and viruel when a is the unique abelian subgroup of index p in s and is not essential equivalently is not radical in the fusion system we give a complete list of all possibilities which can occur this includes several families of exotic fusion systems including some which have proper strongly closed subgroups

zj controlled blocks of quasisimple groups for odd primes abstract we investigated the converses of glauberman s zj theorem and its block theoretic analogue proved by kessar linckelmann and robinson for a finite group g we proved that a block b of g with a defect group d is zj controlled that is n g z j d controls b fusion in g if and only if b is controlled provided that g is quasisimple and the characteristic p of the modular representation is odd this implies by a result of 5 that the converses of glauberman s zj theorem and its block theoretic analogue both hold in these cases as an application we obtain all q d p free quasisimple groups

entry times distribution for mixing systems we consider the return times dynamics to bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties these mixing properties are satisfied for instance by systems that allow young tower constructions we show that the higher order return times to bowen balls are in the limit poisson distributed we also provide a general result for the asymptotic behavior of the recurrence time for bowen balls for ergodic systems and those with specification

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

sunklodas approach to normal approximation for time dependent dynamical systems we consider time dependent dynamical systems arising as sequential compositions of self maps of a probability space we establish conditions under which the birkhoff sums for multivariate observations given a centering and a general normalizing sequence b n of invertible square matrices are approximated by a normal distribution with respect to a metric of regular test functions depending on the metric and the normalizing sequence b n the conditions imply that the error in the approximation decays either at the rate o n 1 2 or the rate o n 1 2 log n under the additional assumption that vert b n 1 vert lesssim n 1 2 the error comes with a multiplicative constant whose exact value can be computed directly from the conditions the proof is based on an observation due to sunklodas regarding stein s method of normal approximation we give applications to one dimensional random piecewise expanding maps and to sequential random and quasistatic intermittent systems

a characterization of solvability for finite groups in terms of their frame abstract the frame of a group is the poset of conjugacy classes of all its proper subgroups in this paper we will prove that a finite group is solvable if and only if every collection of maximal elements of its frame has a well defined meet and the poset consisting of all such meets including the meet of the empty set is a modular lattice

controlled 2 blocks of quasisimple groups abstract controlled p blocks introduced by alperin and broue in 1979 were classified for quasisimple groups g for odd primes in 10 we classify the controlled 2 blocks of quasisimple groups g the results imply that every nilpotent 2 block of g has abelian defect groups this agrees with one of the main results proved in 15 we also give an explicit characterization of non controlled 2 blocks of all quasisimple groups this implies that the p 2 version of the block theoretic analogue of glauberman s zj theorem proved by kessar linckelmann and robinson in 27 holds for quasisimple groups g

from thompson to baer suzuki a sharp characterization of the solvable radical abstract we prove that an element g of prime order 3 belongs to the solvable radical r g of a finite or more generally a linear group if and only if for every x g the subgroup generated by g x g x 1 is solvable this theorem implies that a finite or a linear group g is solvable if and only if in each conjugacy class of g every two elements generate a solvable subgroup

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

on conjugacy classes of maximal subgroups of finite simple groups and a related zeta function we prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of lie type of bounded rank is bounded for exceptional groups this solves a longstanding open problem the proof uses among other tools some methods from geometric invariant theory using this result we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of lie type groups of fixed rank drawing conclusions regarding the behaviour of the corresponding zeta function g s m maxg g m s which appears in many probabilistic applications more specifically we are able to show that for simple groups g and for any fixed real number s 1 g s 0 as g this confirms a conjecture made in 27 we also apply these results to prove the conjecture made in 29 that the symmetric group sn has n o 1 conjugacy classes of primitive maximal subgroups 2000 mathematics classification numbers 20e28 20g15 20d06

a solvable version of the baer suzuki theorem suppose that g is a finite group and x g has prime order p 5 then x is contained in the solvable radical of g o g if and only if x x g is solvable for all g g if g is an almost simple group and x g has prime order p 5 then this implies that there exists g g such that x x g is not solvable in fact this is also true when p 3 with very few exceptions which are described explicitly

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

all p local finite groups of rank two for odd prime p in this paper we give a classification of the rank two p local finite groups for odd p this study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible f radical subgroups also for each case in the classification either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p local finite group getting some new examples of these for p 3

feedback stabilization of a class of nonsmooth systems in this paper we investigate global stabilization of nonsmooth systems the aim is to provide sufficient condition based on standard derivative for feedback stabilization of nonsmooth systems by means of a feedback of class cr r 1 the main result is a generalization of a result concerning smooth systems see for instance j control optimization 32 1994 358 to systems that are only continuous and locally lipschitz

subgroups of type a1 containing a fixed unipotent element in an algebraic group keywords simple algebraic groups unipotent elements subgroups of type a 1 frobenius morphisms graph morphisms finite groups of lie type reference ctg article 2000 003doi 10 1006 jabr 2000 8353 record created on 2008 12 16 modified on 2017 05 12

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

a simple construction of the ree groups of type 2f4 abstract we give an elementary construction of ree s family of finite simple groups f 4 2 q avoiding the need for the machinery of lie algebras algebraic groups or buildings we calculate the group orders and prove simplicity from first principles moreover this is a practical construction in the sense that it gives an explicit description of the generalized octagon and from it generators for many of the maximal subgroups may be easily obtained

products of conjugacy classes in finite and algebraic simple groups abstract we prove the arad herzog conjecture for various families of finite simple groups if a and b are nontrivial conjugacy classes then a b is not a conjugacy class we also prove that if g is a finite simple group of lie type and a and b are nontrivial conjugacy classes either both semisimple or both unipotent then a b is not a conjugacy class we also prove a strong version of the arad herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes as a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product a special case of this has been used by prasad to prove a uniqueness result for tits systems in quasi reductive groups our final result is a generalization of the baer suzuki theorem for p elements with p 5

stabilizability of the angular velocity of a rigid body revisited abstract we prove that the jurdjevic quinn techniques for smooth stabilization can be applied to the equation of the angular velocity of a rigid body an explicit polynomial control is given for the symmetric case

linearized control systems and applications to smooth stabilization for a control system dot x f x u the author proves that for generic feedback laws u such that f x u x does not vanish the linearized control systems around the trajectories of dot x f x u x have the same strong accessibility algebra as f applications are given to the smooth stabilization problem

primitive permutation groups with a solvable 2 transitive subconstituent for a permutation group g acting on a finite set and a point a suborbit is an orbit of the point stabilizer g on the permutation group induced by g on is called a subconstituent of g moreover g is said to be uniprimitive if g is primitive but not 2 transitive in this paper we investigate uniprimitive permutation groups which have a solvable 2 transitive subconstituent we determine all such groups g which have a simple socle the affine case that is g has an elementary abelian socle are also discussed and an infinite family of affine primitive groups with non self paired 2 transitive subconstituents are presented

stabilizability of closed orbits abstract the main contribution of this paper is a version of the jurdjevic quinn method for feedback stabilization of closed orbits of autonomous affine control systems

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

asymptotics for hitting times essentially decreasing sequences of balls in ametric space x and identi ed to be the distribution function of the posi tive exponential law with parameter 1 in many classes of mixing systems in abadi and galves 2001 collet and galves 1993 hirata saussol andvaienti 1999 saussol 1998 and young 1999 nonexponential asymp totics have been obtained in coelho and de faria 1990 for instance theliterature on the subject is rather important and our list is incomplete for further information we refer the reader to the surveys abadi 2004 orcoelho 2000 there exists a connection between return time asymptotics and hittingtime asymptotics indeed as shown in hirata saussol and vaienti 1999 when the asymptotics for return times is exponential with parameter 1 thenso is the one for hitting times possible asymptotics for return times that is weak limits for f

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

controlled blocks of the finite quasisimple groups for odd primes abstract we classify controlled blocks introduced by alperin and broue in 1979 for all quasisimple groups g for odd primes the results imply that every nilpotent block of g has abelian defect groups which in turn is one of the main results proved in an and eaton 2011 6 we also give an explicit characterization of non controlled blocks of all quasisimple groups g for odd primes this implies the block theoretic analogue of glauberman s zj theorem for g proved by kessar linckelmann and robinson 2002 18

transitive subgroups of primitive permutation groups abstract we investigate the finite primitive permutation groups g which have a transitive subgroup containing no nontrivial subnormal subgroup of g the conclusion is that such primitive groups are rather rare and that their existence is intimately connected with factorisations of almost simple groups a corollary is obtained on primitive groups which contain a regular subgroup heavily involved in our proofs are some new results on subgroups of simple groups which have orders divisible by various primes for example another corollary implies that for every simple group t apart from l 3 3 u 3 3 and l 2 p with p a mersenne prime there is a collection consisting of two or three odd prime divisors of t such that if m is a subgroup of t of order divisible by every prime in then m is divisible by all the prime divisors of t and we obtain a classification of such subgroups m

recognizing the characteristic of a simple group of lie type from its probabilistic zeta function abstract we find a method to recognize the characteristic of a simple group of lie type g from its dirichlet polynomial p g s this is enough to complete the proof of the following statement if g is a simple group h is a finite group and p g s p h s then h frat h g

feedback stabilization of general nonlinear control systems a passive system approach abstract this paper develops sufficient conditions for a general nonlinear control system 1 x f x u to be locally resp globally asymptotically stabilizable via smooth state feedback in particular it is shown that as in the case of affine systems this is possible if the unforced dynamic system of 1 is lyapunov stable and appropriate controllability like rank conditions are satisfied our results incorporate a series of well known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems

affine planes admitting a collineation group with a transitive action on the line at infinity abstract the affine planes admitting a collineation group with a transitive action on the line at infinity are investigated and an essentially complete classification is achieved when the group is almost simple a rather deep result is also obtained when the action on the line at infinity is faithful and primitive

the alperin and dade conjectures for ree groups2f4 q2 in non defining characteristics abstract this paper is part of a program to study alperin s weight conjecture and dade s ordinary conjecture on counting characters in blocks for several finite groups the local structures of radical subgroups and certain radical 3 chains of a ree group of typef4are given and the conjectures and a conjecture of olsson are confirmed for the group when the characteristic of the modular representation is distinct from the defining characteristic of the group

the simple ree groups f42 q2 are determined by the set of their character degrees let g be a finite group let cd g be the set of all complex irreducible character degrees of g in this paper we will show that if cd g cd h where h is the simple ree group f42 q2 q2 8 then g h a where a is an abelian group this verifies huppert s conjecture for the simple ree groups f42 q2 when q2 8

bounded smooth state feedback and a global separation principle for non affine nonlinear systems abstract in this paper we show that a single input nonlinear system z 0 x g 1 x u g l x u l is globally asymptotically stabilizable by arbitrarily small smooth state feedback if unforced dynamics of are lyapunov stable and satisfies appropriate controllability like rank conditions characterized by the lie bracket of vector fields f0 x and g1 x an explicit smooth state feedback control law is constructed this in turn leads to an analytic resp smooth state feedback control law which solves the problem of global stabilization for analytic resp smooth systems x f x u based on these results we then establish a global separation principle for the system with an output finally we briefly discuss how the results can be extended to a multi input system

on feedback stabilization of smooth nonlinear systems this paper deals with the problem of stabilization of nonlinear systems we present a version of the well known jurdjevic quinn theorem 1978 concerning affine systems with dissipative drift for smooth but not necessarily analytic systems an example and counterexample are proposed

recurrence dimensions and lyapunov exponents we show that the poincare return time of a typical cylinder is at least its length for one dimensional maps we express the lyapunov exponent and dimension via return times

hitting and return times in ergodic dynamical systems given an ergodic dynamical system x t and u c x measurable with u 0 let u u x denote the normalized hitting time of x x to u we prove that given a sequence u n with u n 0 the distribution function of the normalized hitting times to u n converges weakly to some subprobability distribution f if and only if the distribution function of the normalized return time converges weakly to some distribution function f and that in the converging case f t 0 t 1 f s ds t 0 this in particular characterizes asymptotics for hitting times and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also

further solvable analogues of the baer suzuki theorem and generation of nonsolvable groups abstract let g be an almost simple group we prove that if x g has prime order p 5 then there exists an involution y such that x y is not solvable also if x is an involution then there exist three conjugates of x that generate a nonsolvable group unless x belongs to a short list of exceptions which are described explicitly we also prove that if x has order 6 or 9 then there exist two conjugates that generate a nonsolvable group

automorphisms of k groups i abstract this is the first in a sequence of papers that will develop the theory of automorphisms of nonsolvable finite groups the sequence will culminate in a new proof of mcbride s nonsolvable signalizer functor theorem which is one of the fundamental results required for the proof of the classification of the finite simple groups

on asymptotically stabilizing feedback control of bilinear systems asymptotically stabilizing feedback controls for a class of bilinear systems are discussed a simple modification which strengthens a result of slemrod 1 is noted and illustrated by example