generation and random generation from simple groups to maximal subgroups abstract let g be a finite group and let d g be the minimal number of generators for g it is well known that d g 2 for all non abelian finite simple groups we prove that d h 4 for any maximal subgroup h of a finite simple group and that this bound is best possible we also investigate the random generation of maximal subgroups of simple and almost simple groups by applying a recent theorem of jaikin zapirain and pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant we then apply our results to the study of permutation groups in particular we show that if g is a finite primitive permutation group with point stabilizer h then d g 1 d h d g 4

on the product of a nilpotent group and a group with non trivial center abstract it is proved that a finite group g a b which is a product of a nilpotent subgroup a and a subgroup b with non trivial center contains a non trivial abelian normal subgroup

some locally 3 arc transitive graphs constructed from triality abstract two new infinite families of locally 3 arc transitive graphs are constructed which have aut p 8 q as their automorphism group

on classifying all full factorisations and multiple factorisations of the finite almost simple groups abstract we present some general results on factorisations of almost simple groups these results are consequences of the classification of all maximal factorisations of almost simple groups due to liebeck saxl and the second author and are needed for applications to the theory of quasiprimitive permutation groups

double low rank matrix recovery for saliency fusion in this paper we address the problem of fusing various saliency detection methods such that the fusion result outperforms each of the individual methods we observe that the saliency regions shown in different saliency maps are with high probability covering parts of the salient object with image regions being represented by the saliency values of multiple saliency maps the object regions have strong correlation and thus lie in a low dimensional subspace meanwhile most of the background regions tend to have lower saliency values in various saliency maps they are also strongly correlated and lie in a low dimensional subspace that is independent of the object subspace therefore an image can be represented as the combination of two low rank matrices to obtain a unified low rank matrix that represents the salient object this paper presents a double low rank matrix recovery model for saliency fusion the inference process is formulated as a constrained nuclear norm minimization problem which is convex and can be solved efficiently with the alternating direction method of multipliers furthermore to reduce the computational complexity of the proposed saliency fusion method a saliency model selection strategy based on the sparse representation is proposed experiments on five datasets show that our method consistently outperforms each individual saliency detection approach and other state of the art saliency fusion methods

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

on representing finite lattices as intervals in subgroup lattices of finite groups let mnbe the lattice of length 2 withn 1 atoms it is an open problem to decide whether or not every such lattice or indeed whether or not every finite lattice can be represented as an interval in the subgroup lattice of some finite group we complete the work of the second author lucchini by reducing this problem to a series of questions concerning the finite non abelian simple groups

generation of finite classical groups by pairs of elements with large fixed point spaces abstract we study good elements in finite 2 n dimensional classical groups g namely t is a good element if o t is divisible by a primitive prime divisor of q n 1 for the relevant field order q and t fixes pointwise an n space the group sl 2 n q contains such elements and they are present in su 2 n q sp 2 n q so 2 n e q only if n is odd even even respectively we prove that there is an absolute positive constant c such that two random conjugates of t generate g with probability at least c if g so 2 n e 2 and g sp 2 n q with q even in the exceptional case g sp 2 n q with q even two conjugates of t never generate g in this case we prove that two random conjugates of t generate a subgroup so 2 n e q with probability at least c the results underpin analysis of new constructive recognition algorithms for classical groups in even characteristic which succeed where methods utilising involution centralisers are not available

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

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

cyclic quotients of transitive groups abstract let a be a transitive subgroup of s n we show that the largest cyclic quotient of a has order at most n this can be interpreted as an equivalent result about extensions of constants in the galois closure of a covering of curves over a finite field we also prove that the point stabilizer in a finite primitive permutation group always has a faithful orbit

rouquier blocks in chevalley groups of type e abstract in finite chevalley groups of type e we shall find a rouquier like block algebra in non defining characteristic we construct a morita equivalence between this block and a principal block of a local subgroup as an application of the equivalence we shall determine some decomposition numbers in type e

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

primitive permutation groups of bounded orbital diameter we give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs this is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive a key result is that in the almost simple case with socle of fixed lie rank apart from very specific cases there is such a diameter bound this is proved using recent results on the model theory of pseudofinite fields and difference fields

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

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

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

the probabilistic zeta function of finite simple groups abstract let g be a finite group there exists a uniquely determined dirichlet polynomial p g s such that if t n then p g t gives the probability of generating g with t randomly chosen elements we show that it may be recognized from the knowledge of p g s whether g frat g is a simple group

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

primitive permutation groups with a regular subgroup abstract this paper starts the classification of the primitive permutation groups g such that g contains a regular subgroup x we determine all the triples g x with soc g an alternating or a sporadic or an exceptional group of lie type further we construct all the examples g x with g a classical group which are known to us our particular interest is in the 8 dimensional orthogonal groups of witt index 4 we determine all the triples g x with soc g p 8 q in order to obtain all these triples we also study the almost simple groups g with g p 2 n 1 q the case g u n q is started in this paper and finished in b baumeister primitive permutation groups of unitary type with a regular subgroup bull belg math soc 112 5 2006 657 673 a group x is called a burnside group or short a b group if each primitive permutation group which contains a regular subgroup isomorphic to x is necessarily 2 transitive in the end of the paper we discuss b groups

fixed point ratios in actions of finite classical groups iii abstract this is the first in a series of four papers on fixed point ratios in actions of finite classical groups our main result states that if g is a finite almost simple classical group and is a faithful transitive non subspace g set then either fpr x x g 1 2 for all elements x g of prime order or g is one of a small number of known exceptions here fpr x denotes the proportion of points in which are fixed by x in this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups a further application concerning monodromy groups of covers of riemann surfaces is also outlined

sylow s theorem for moufang loops abstract for finite moufang loops we prove an analog of the first sylow theorem giving a criterion for the existence of a p sylow subloop we also find the maximal order of p subloops in the moufang loops that do not possess p sylow subloops