finite permutation groups with a transitive minimal normal subgroup a finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup in this paper we give a characterisation and structure theorem for the finite innately transitive groups as well as describing those innately transitive groups which preserve a product decomposition the class of innately transitive groups contains all primitive and quasiprimitive groups

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

regularity results for real analytic homotopies in this paper we study two main features of the homotopy curves which we follow when we use the homotopy method for solving the zeros of analytic maps first we prove that near the solution the curve behaves nicely secondly we prove that the set of starting points which give smooth homotopy curves is open and dense the second property is of particular importance in computer implementation

regularity results for solving systems of polynomials by homotopy method the homotopy method for solving systems of polynomial equations proposed by li also morgan is believed to be one of the simplest for computer implementation it is shown here that the parameters of the homotopy can be chosen from an open and dense set such that all isolated solutions of polynomial systems can be obtained

the factorizations of the finite exceptional groups of lie type a finite group is said to be factorizable if g ab for some proper subgroups a and b of g such an expression is called a factorization of g factorizations have been the subject of considerable study see for example 11 121 for the case where g is a finite simple group some results can be found in 8 20 26 27 291 in this paper we determine completely the factorizations of the exceptional simple groups of lie type and their automorphism groups here by an exceptional simple group of lie type we mean a non abelian finite simple group associated with one of the families g f4 e e e b 2g2 3d f4 and 2e excluding g 2 and g 3 in view of the isomorphisms g 2 e u 3 and g 3 z l 8 there are two interesting features of our main result one is that remarkably few of the exceptional simple groups of lie type are factorizable for example e q ec q e q and e q possess no factorizations in contrast some striking factorizations are found to exist for the groups g q and f q for g q these factorizations are already known 20 271 in another paper 151 the maximal factorizations of all the other finite simple groups and their automorphism groups are classified where the factorization g ab is maximal if both a and b are maximal subgroups of g the results of this paper and 151 are used in 161 to obtain a classification of the maximal subgroups of the finite alternating and symmetric groups our main results here are the following two theorems

dimension raising homomorphisms between lattices of convex bodies abstract we settle the first unsolved case of a problem of p m gruber asked by him in 1991 in 9 namely to investigate the homomorphisms from the lattice of convex bodies of e c to the lattice of convex bodies of e d for c d after establishing dimension bounds for such homomorphism in general we completely describe these homomorphisms for the case d c 1 for c 3 the case d c was settled in 9 the obtained result is then applied to characterise homomorphisms and anti homomorphisms from lattices of convex bodies to lattices of convex functions the relationship between such homomorphisms and anti homomorphisms is furnished by the legendre fenchel transform

the concept of duality in convex analysis and the characterization of the legendre transform in the main theorem of this paper we show that any involution on the class of lower semi continuous convex functions which is order reversing must be up to linear terms the well known legendre transform

the finite primitive permutation groups containing an abelian regular subgroup a complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup this solves a long standing open problem in permutation group theory initiated by w burnside in 1900

quasiprimitive groups and blow up decompositions abstract the blow up construction by l g kovacs has been a very useful tool for studying embeddings of finite primitive permutation groups into wreath products in product action in the present paper we extend the concept of a blow up to finite quasiprimitive permutation groups and use it to study embeddings of finite quasiprimitive groups into wreath products

transitive characteristically simple subgroups of finite quasiprimitive permutation groups abstract the first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product this allows us to characterize nonabelian transitive characteristically simple subgroups h of finite quasiprimitive permutation groups g if the socle of g denoted by soc g is nonabelian then either h is intransitive or soc g is an alternating group in its natural action an explicit description is given for the possibilities of h under the condition that h does not contain a nontrivial normal subgroup of soc g

intransitive cartesian decompositions preserved by innately transitive permutation groups a permutation group is innately transitive if it has a transitive minimal normal subgroup which is referred to as a plinth we study the class of finite innately transitive permutation groups that can be embedded into wreath products in product action this investigation is carried out by observing that such a wreath product preserves a natural cartesian decomposition of the underlying set previously we classified the possible embeddings in the case where the innately transitive group projects onto a transitive subgroup of the top group in this article we prove that the transitivity assumption we made in the previous paper was not too restrictive indeed the image of the projection into the top group can only be intransitive when the finite simple group that is involved in the plinth comes from a small list even then the innately transitive group can have at most three orbits on an invariant cartesian decomposition a consequence of this result is that if g is an innately transitive subgroup of a wreath product in product action then the natural projection of g into the top group has at most two orbits

a characterization of the support map abstract in this short note we give a characterization of the support map from classical convexity we show it is the unique additive transformation from the class of closed convex sets in r n which include 0 to the class of positive 1 homogeneous functions on r n this will be a consequence of a theorem about transforms from the class of convex sets to itself which preserve minkowski addition

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

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

a characterization of duality through section projection correspondence in the finite dimensional setting in this paper we show that the well known duality operation in the context of convex bodies in rn is completely characterized by its property of interchanging sections with projections our results are compared to results by boroczky schneider and artstein milman who showed that in many cases the property of order reversing is sufficient to determine a duality operation up to obvious linear modifications in fact we provide another result that recovers a known characterization of duality by the property of order reversing and up to a mild condition also a characterization of duality by interchanging sections by projections

innately transitive subgroups of wreath products in product action a permutation group is innately transitive if it has a transitive minimal normal subgroup which is referred to as a plinth we study the class of finite innately transitive permutation groups that can be embedded into wreath products in product action this investigation is carried out by observing that such a wreath product preserves a natural cartesian decomposition of the underlying set previously we classified the possible embeddings in the case where the plinth is simple here we extend that classification and identify several different types of cartesian decompositions that can be preserved by an innately transitive group with a non abelian plinth these different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action we also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products

on finite permutation groups with a transitive cyclic subgroup abstract in this paper we study the structure of finite permutation groups with a transitive cyclic subgroup in particular we extend the classification due to w feit and g a jones of the primitive groups containing such subgroups to permutation groups that are i quasiprimitive ii almost simple and iii innately transitive we also analyse the actions of normal subgroups of arbitrary groups with transitive cyclic subgroups we give an application to circulant homogeneous factorisations of complete graphs obtaining new structural information and giving a new proof of their possible parameters

on primitive overgroups of quasiprimitive permutation groups a permutation group is said to be quasiprimitive if all its non trivial normal subgroups are transitive we investigate pairs g h of permutation groups of degree n such that g h sn with g quasiprimitive and h primitive an explicit classification of such pairs is obtained except in the cases where the primitive group h is either almost simple or the blow up of an almost simple group the theory in these remaining cases is investigated in separate papers the results depend on the finite simple group classification

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

factorisations of characteristically simple groups abstract we solve some factorisation problems for finite characteristically simple groups using factorisations of elementary abelian 2 groups the problems arose from a study of permutation groups we found the connection between the factorisations of these two types of groups surprisingly interesting

an approximate solution treatment for non linear problems in this paper we study one simple typical non linear equation to describe a kind of analytical technique for non linear problems this technique is based on both homotopy in topology and the maclaurin series in contrast to perturbation techniques the homotopy technique does not require small or large parameters the results show that the homotopy technique can give much better approximation than given by perturbation techniques in addition the homotopy technique can be used to obtain formula uniformly valid for both small and large parameters in non linear problems it is clear that the homotopy technique is suitable for strongly non linear problems