on the multiplicity free actions of the sporadic simple groups abstract we introduce a database containing the character tables of the endomorphism rings of the multiplicity free permutation modules of the sporadic simple groups their automorphism groups their schur covers and their bicyclic extensions we describe the techniques used to compile the data and present a couple of applications to orbital graphs

a two dimensional version of the goldschmidt sims conjecture abstract the goldschmidt sims conjecture asserts that there is a finite number of conjugacy classes of edge transitive lattices in the automorphism group of a regular tree with prime valence we prove a similar theorem for irreducible lattices transitive on the 2 cells of the product of two regular trees of prime valences

on the order of borel subgroups of group amalgams and an application to locally transitive graphs abstract a permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular given nontrivial finite transitive permutation groups l 1 and l 2 with l 1 not semiprimitive we construct an infinite family of rank two amalgams of permutation type l 1 l 2 and borel subgroups of strictly increasing order as an application we show that there is no bound on the order of edge stabilisers in locally l 1 l 2 graphs we also consider the corresponding question for amalgams of rank k 3 we completely resolve this by showing that the order of the borel subgroup is bounded by the permutation type l 1 l k only in the trivial case where each of l 1 l k is regular

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 the automorphism groups of quasiprimitive almost simple graphs let gamma be a graph and let g be a subgroup of automorphisms of gamma then g is said to be locally primitive on gamma if for each vertex upsilon the stabilizer g upsilon induces a primitive group of permutations on the set of vertices adjacent to upsilon this paper investigates pairs gamma g for which g is locally primitive on gamma g is an almost simple group that is l less than or equal to g less than or equal to aut l for some nonabelian simple group l and the simple socle l is transitive on vertices each such graph is a cover of a possibly smaller graph l gamma over tildeg on which g is also locally primitive and for which in addition aut l gamma over tildeg is quasiprimitive on vertices that is every nontrivial normal subgroup of aut l gamma over tildeg is vertex transitive it is proved that aut l gamma over tildeg is also an almost simple group tn the general case in which aut gamma is not quasiprimitive on vertices we show that either every intransitive minimal normal subgroup of aut gamma centralizes l or l is of lie type and aut gamma involves an explicitly known same characteristic module for l c 1999 academic press

on locally semiprimitive graphs and a theorem of weiss abstract in this paper we investigate graphs that admit a group acting arc transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup this type of semiprimitive group is a generalisation of an affine group we show that if the graph has valency coprime to six then there is a bound on the order of the vertex stabilisers depending on the valency alone we also prove a detailed structure theorem for the vertex stabilisers in the remaining case this is a contribution to an ongoing project to investigate the validity of the potocnik spiga verret conjecture

on groups with every normal subgroup transitive or semiregular we investigate finite solvable permutation groups in which all normal subgroups are transitive or semiregular the motivation comes from universal algebra such groups are examples of collapsing transformation monoids

on the orders of primitive groups abstract almost all primitive permutation groups of degree n have order at most n i 0 log 2 n 1 n 2 i 1 log 2 n or have socle isomorphic to a direct power of some alternating group the mathieu groups m11 m12 m23 and m24 are the four exceptions as a corollary the sharp version of a theorem of praeger and saxl is established where m12 turns out to be the largest primitive group for an application a bound on the orders of permutation groups without large alternating composition factors is given this sharpens a lemma of babai cameron palfy and generalizes a theorem of dixon

distance transitive graphs and finite simple groups introductionthis paper represents the first step in the classification of finite primitive distancetransitive graphs in it we reduce the problem to the case where the automorph ism group is either almost simple or affine let be a simple connected undirected graph with verte oc x se t q e q i fthen d a j8 denotes the distance between 3 a i ann

on the order of uniprimitive permutation groups one of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group g of degree n where g x an we prove i g i exp 4v n log2 n for the case when g is not doubly transitive this result is best possible apart from an o log n factor in the exponent the best result previously known was i g i el h wielandt a similar estimate for the doubly transitive case follows in a subsequent paper for rank 3 groups with subdegree p n 2 we obtain i gi exp 4 n p log2 n in the proof we develop some new combinatorial properties of coherent configurations the results also have relevance to theoretical estimates on the computational complexity of graph isomorphism testing

a classification of primitive permutation groups with finite stabilizers abstract we classify all infinite primitive permutation groups possessing a finite point stabilizer thus extending the seminal aschbacher o nan scott theorem to all primitive permutation groups with finite point stabilizers

the hardy class of geometric models and the essential spectral radius of composition operators let be an analytic map of the disk into itself that fixes the origin then it is well known that induces a bounded composition operatorc on the hardy spaceh2 d also by a classical result of k nigs there is an analytic map such that 0 0 which is one to one when is one to one recently p bourdon and j shapiro found a lower bound for the essential spectral radius ofc in terms of andh sup p 0 hp d and asked for an exact formula in the following we assume that is one to one we determine the hardy class of in terms of the image setg d and then show that bourdon and shapiro s lower bound is also an upper bound hence answering their question in the case of univalent symbols moreover in an earlier paper shapiro smith and stegenga introduced the concept of twisted sectors to estimateh and we showed p poggi corradini trans amer math soc 348 6 1996 2503 2518 how the dynamics of the map and some harmonic measure estimates allow one to completely characterize the caseh in terms of twisted sectors by further analyzing this concept we produce a new characterization ofh in terms of the invariant sector like regions that are contained ing d this different approach also yields another proof of bourdon and shapiro s lower bound mentioned above in our specific case

on pyber s base size conjecture let g be a permutation group on a finite set a subset of is a base for g if its pointwise stabilizer in g is trivial the base size of g denoted b g is the smallest size of a base a well known conjecture of pyber from the early 1990s asserts that there exists an absolute constant c such that b g c log g logn for any primitive permutation group g of degree n several special cases have been verified in recent years including the almost simple and diagonal cases in this paper we prove pyber s conjecture for all non affine primitive groups

a theory of semiprimitive groups abstract a transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular interest in this class of groups is motivated by two sources problems arising in universal algebra related to collapsing monoids and the graph restrictive problem for permutation groups here we develop a theory of semiprimitive groups which encompasses their structure their quotient actions and a method by which all finite semiprimitive groups are constructed we also extend some results from the theory of primitive groups to semiprimitive groups and conclude with open problems of a similar nature

amalgams of prime index this serves as a motivation for our main theorem from the theory of the classification of finite simple groups indeed our result classifies all such configurations and in particular it says that in all the interesting cases we must have ip n p i d 2 and ip p n p i 3 or 5 the basic definitions we use are the same as in gs which we quickly repeat for the convenience of the reader an umalgatn consists of two monomorphisms 4 and dz of groups such that

geometric models and compactness of composition operators abstract this work explores some of the terrain between functional equations geometric function theory and operator theory it is inspired by the fact that whenever a composition operator or one of its powers is compact on the hardy space h 2 then its eigenfunctions cannot grow too quickly on the unit disc the goal is to show that under certain natural and necessary additional conditions there is a converse slow growth of eigenfunctions implies compactness we interpret the slow growth condition in terms of the geometry of the principal eigenfunction of the composition operator the konigs function of the inducing map we emphasize throughout the importance of this eigenfunction in providing a simple geometric model for the operator s inducing map

mean growth of koenigs eigenfunctions in 1884 g koenigs solved schroeder s functional equation f o a af in the following context is a given holomorphic function mapping the open unit disk u into itself and fixing a point a e u f is holomorphic on u and a is a complex scalar koenigs showed that if 0 equation for p has a unique holomorphic solution cr satisfying a co p a a and o 0 1 moreover he showed that the only other solutions are the obvious ones given by constant multiples of powers of a we call a the koenigs eigenfunction of motivated by fundamental issues in operator theory and function theory we seek to understand the growth of integral means of koenigs eigenfunctions for 0 of to belong to the hardy space hp and show that the condition is necessary when is analytic on the closed disk for many mappings the condition may be expressed as a relationship between p a and derivatives of at points on ou that are fixed by some iterate of our work depends upon a formula we establish for the essential spectral radius of any composition operator on the hardy space hp department of mathematics washington and lee university lexington virginia 24450 e mail address pbourdonfwlu edu department of mathematics michigan state university east lansing michigan 48824 e mail address shapirofmath msu edu this content downloaded from 157 55 39 211 on sat 25 jun 2016 06 33 26 utc all use subject to http about jstor org terms

hardy spaces and twisted sectors for geometric models we study the one to one analytic maps a that send the unit disc into a region g with the property that ag c g for some complex number a 0 jai 1 these functions arise in iteration theory giving a model for the self maps of the unit disk into itself and in the study of composition operators as their eigenfunctions we show that for such functions there are geometrical conditions on the image region g that characterize their rate of growth i e we prove that a egnp hp if and only if g does not contain a twisted sector then we examine the connection with composition operators and further investigate the no twisted sector condition finally in the appendix we give a different proof of a result of j shapiro about the essential norm of a composition operator

normalizers of primitive permutation groups abstract let g be a transitive normal subgroup of a permutation group a of finite degree n the factor group a g can be considered as a certain galois group and one would like to bound its size one of the results of the paper is that a g n if g is primitive unless n 3 4 54 38 58 or 316 this bound is sharp when n is prime in fact when g is primitive out g n unless g is a member of a given infinite sequence of primitive groups and n is different from the previously listed integers many other results of this flavor are established not only for permutation groups but also for linear groups and galois groups