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Let Γ be a discrete group. A C*-algebra A is an exotic C*-algebra (associated to Γ) if there exist proper surjective C*-quotients C ⁎ ( Γ ) → A → C r ⁎ ( Γ ) which compose to the canonical quotient C ⁎ ( Γ ) → C r ⁎ ( Γ ) . In this paper, we show that a large class of exotic C*-algebras has poor local properties. More precisely, we demonstrate the failure of local reflexivity, exactness, and local lifting property. Additionally, A does not admit an amenable trace and, hence, is not quasidiagonal and does not have the WEP when A is from the class of exotic C*-algebras defined by Brown and Guentner (see [8] ). In order to achieve the main results of this paper, we prove a result which implies the factorization property for the class of discrete groups which are algebraic subgroups of locally compact amenable groups. [ABSTRACT FROM AUTHOR]

Hooks are prominent in representation theory (of symmetric groups) and they play a role in number theory (via cranks associated to Ramanujan's congruences). A partition of a positive integer n has a Young diagram representation. To each cell in the diagram there is an associated statistic called hook length, and if a number t is absent from the diagram then the partition is called a t -core. A partition is an ( s , t ) -core if it is both an s - and a t -core. Since the work of Anderson on ( s , t ) -cores, the topic has received growing attention. This paper expands the discussion to multiple-cores. More precisely, we explore ( s , s + 1 , … , s + k ) -core partitions much in the spirit of a recent paper by Stanley and Zanello. In fact, our results exploit connections between three combinatorial objects: multi-cores, posets and lattice paths (with a novel generalization of Dyck paths). Additional results and conjectures are scattered throughout the paper. For example, one of these statements implies a curious symmetry for twin-coprime ( s , s + 2 ) -core partitions. [ABSTRACT FROM AUTHOR]

We associate with every group G a set τ ( G ) of subgroups of G with 1 ∈ τ ( G ) . If H ∈ τ ( G ) , then we say that H is a τ-subgroup of G . If θ ( τ ( G ) ) ⊆ τ ( θ ( G ) ) for each epimorphism θ : G → G ⁎ , then we say that τ is a subgroup functor . We say also that a subgroup functor τ is: hereditary provided H ∈ τ ( E ) whenever H ≤ E ≤ G and H ∈ τ ( G ) ; regular provided for any group G , whenever H ∈ τ ( G ) is a p -group and N is a minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p ; Φ -regular (respectively Φ -quasiregular ) provided for any primitive group G , whenever H ∈ τ ( G ) is a p -group and N is a (respectively abelian) minimal normal subgroup of G , then | G : N G ( H ∩ N ) | is a power of p . Let K ≤ H be subgroups of G and τ a subgroup functor. Then we say that: the pair ( K , H ) satisfies the F -supplement condition in G if G has a subgroup T such that H T = G and H ∩ T ⊆ K Z F ( T ) ; H is F τ -supplemented in G if for some τ -subgroup S ¯ of G ¯ contained in H ¯ the pair ( S ¯ , H ¯ ) satisfies the F -supplement condition in G ¯ , where G ¯ = G / H G and H ¯ = H / H G . In this paper we study the structure of a group G under the condition that some primary subgroups of G are F τ -supplemented in G . In particular, we prove the following result. Theorem A. Let F be a saturated formation containing the class U of all supersoluble groups, E a normal subgroup of G with G / E ∈ F , X = E or X = F ⁎ ( E ) , and τ a regular or hereditary Φ -regular subgroup functor. Suppose that every τ-subgroup of G contained in X is subnormally embedded in G. If every maximal subgroup of every non-cyclic Sylow subgroup of X is U τ -supplemented in G, then G ∈ F . Moreover, in the case when τ is regular, then every chief factor of G below E is cyclic. The results in this paper not only cover and unify a long list of some known results but also cause a wide series of new results. [ABSTRACT FROM AUTHOR]

We consider a group-theoretic analogue of the classic subset sum problem. It is known that every virtually nilpotent group has polynomial time decidable subset sum problem. In this paper we use subgroup distortion to show that every polycyclic non-virtually-nilpotent group has NP -complete subset sum problem. [ABSTRACT FROM AUTHOR]

A relational structure is homogeneous if every isomorphism between finite induced substructures has an extension to an automorphism of . A relational structure is indivisible if for every partition of the set of elements of , there is a copy of in or in . The structure is weakly indivisible if for every partition of the set of elements of for which some finite induced substructure of does not have a copy in , there exists a copy of in . The structure is age indivisible if for every partition of the elements of every finite induced substructure of has an embedding into or every finite induced substructure of has an embedding into . It follows that indivisibility implies weak indivisibility which in turn implies age indivisibility. There are many examples of countable infinite homogeneous structures which are weakly indivisible but not indivisible; see Sauer (2000) [17] and the end of Section 3.1 of this paper. In general, it is difficult to find structures which are age indivisible but not weakly indivisible. One such example (see Pouzet et al. (2011) [16]) is obtained via the inhomogeneous general linear group of vector spaces over finite fields. Finding such an example for homogeneous structures seems to be even more difficult. The only one that I obtained, together with L. Nguyen Van Thé, is a countable homogeneous metric subspace of the Hilbert sphere (see Nguyen Van Thé and Sauer (2010) [13]). Both of the examples above are relational structures with infinitely many relations. One of the still open questions then is: Are there countable infinite homogeneous structures with finite signature which are age but not weakly indivisible? We will show that the generic “free amalgamation” homogeneous structures are weakly indivisible; see Section 3.1. Countable “Urysohn metric spaces” (see the next section), with finite distance sets, are indivisible (see Sauer (2013) [19]). Countable Urysohn metric spaces are age indivisible, which follows from the Hales–Jewett theorem (Hales and Jewett (1963) [7]; see Delhommé et al. (2007) [2]). For non-Urysohn homogeneous countable metric spaces, except for the example mentioned above, the weak indivisibility question is completely open. We will show, in this paper, that a countable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible. [ABSTRACT FROM AUTHOR]

Abstract: In several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named -groups. Groups with all subgroups being -groups are called groups with the basis property. In the first part of this paper we correct and generalize some characterizations of groups with the basis property. In the second part we consider groups satisfying analogous conditions, but only for sets of generators of prime power orders. The class of groups introduced here is much larger then the class of groups with the basis property. For example, it contains all nilpotent groups. [Copyright &y& Elsevier]

Abstract: Given a subroup G of a stable (in the model-theoric sense) group Γ, in particular when Γ is a group of finite Morley rank, the traces on G of the definable subsets of Γ have a remarkable property: if the definable closure of G is connected, they are either supergeneric, or supergenerically complemented, in the sense of the definition given at the very beginning of this paper. An example of this situation is provided by the linear groups: for some n, G is a subgroup of , where K is a field that we may take algebraically closed; the definable sets in the sense of are its constructible subsets, i.e. the boolean combinations of a finite number of its Zariski closed subsets. For any group G, the supergeneric subsets of G form a filter of large sets, which, to my best knowledge, is defined here for the first time. This paper undertakes the study of supergenericity in a general context, with no hypotheses of a model-theoric nature, but with a special attention given to the very specific properties of genericity possessed by the definable subsets of a stable group. It can be read without any knowledge of Logic, provided that one is ready to skip the proofs of the theorems showing precisely that these definable sets have these properties. [Copyright &y& Elsevier]

Abstract: A graph is called locally-primitive if the vertex stabilizer is primitive on the neighbor set of for each vertex . In this paper, we classify locally-primitive Cayley graphs of dihedral groups, while 2-arc-transitive Cayley graphs of dihedral groups have been classified by a series of papers in the literature. [Copyright &y& Elsevier]

Abstract: Let X be a finite set such that . Let and denote the transformation monoid and the symmetric group on n points, respectively. Given , we say that a group is a-normalizing if where and denote the subsemigroups of generated by the sets and , respectively. If G is a-normalizing for all , then we say that G is normalizing. The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory. [Copyright &y& Elsevier]

For i = 1 , 2 , 3 , let φ i be Young functions, ( Ω , μ ) a (topological) measure space, E an ideal of μ -measurable complex-valued functions defined on Ω and E φ i be the corresponding Calderón–Lozanowskiĭ space. Our aim in this paper is to give, under mild conditions, several results on topological size (in the sense of Baire) of the sets { ( f , g ) ∈ E φ 1 × E φ 2 : | f | ⨀ | g | ∈ E φ 3 } and { ( f , g ) ∈ E φ 1 × E φ 2 : ∃ x ∈ V , ( f ⨀ g ) ( x ) is well defined } where ⨀ denotes the convolution or pointwise product of functions and V a compact neighborhood. Our results sharpen and unify the related results obtained in diverse areas during recent thirty years. [ABSTRACT FROM AUTHOR]

The first part of this paper deals with the combinatorics of equivariant partitions of finite sets with group actions. In the second part, we compute all equivariant Euler characteristics of the Σ n -poset of non-extreme partitions of an n -set. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to define equivariant class group of a locally Krull scheme (that is, a scheme which is locally a prime spectrum of a Krull domain) with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring. In particular, we prove the following. Let k be a field, G a smooth k -group scheme of finite type, and X a quasi-compact quasi-separated locally Krull G -scheme. Assume that there is a k -scheme Z of finite type and a dominant k -morphism Z → X . Let φ : X → Y be a G -invariant morphism such that O Y → ( φ ⁎ O X ) G is an isomorphism. Then Y is locally Krull. If, moreover, Cl ( X ) is finitely generated, then Cl ( G , X ) and Cl ( Y ) are also finitely generated, where Cl ( G , X ) is the equivariant class group. In fact, Cl ( Y ) is a subquotient of Cl ( G , X ) . For actions of connected group schemes on affine schemes, there are similar results of Magid and Waterhouse, but our result also holds for disconnected G . The proof depends on a similar result on (equivariant) Picard groups. [ABSTRACT FROM AUTHOR]

Abstract: Consider a free metabelian Lie algebra M of finite rank r over an infinite field K of prime characteristic p. Given a free generating set, M acquires a grading; its group of graded automorphisms is the general linear group , so each homogeneous component is a finite dimensional -module. The homogeneous component of degree 1 is the natural module, and the other are the metabelian Lie powers of the title. This paper investigates the submodule structure of the . In the main result, a composition series is constructed in each and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of Steinbergʼs twisted tensor product theorem; their dimensions are also given. It turns out that the composition factors are pairwise non-isomorphic, from which it follows that the submodule lattice is finite and distributive. By the Birkhoff representation theorem, any such lattice is explicitly recognizable from the poset of its join-irreducible elements. The poset relevant for is then determined by exploiting a 1975 paper of Yu.A. Bakhturin on identical relations in metabelian Lie algebras. [Copyright &y& Elsevier]

Abstract: Let denote a graph formed from unlabelled graphs and a labelled graph replacing every vertex of by the graph and joining the vertices of with all the vertices of those of whenever . For unlabelled graphs , let stand for the class of all graphs taken over all possible orderings of . A prime inductive class of graphs, , is said to be a set of all graphs, which can be produced by recursive applying of where is a graph from a fixed set of prime graphs and are either graphs from the set of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, -trees, series–parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) Drgas-Burchardt et al. (2010) and Hajós (1961) ). This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class it presents a method for the construction of maximal induced hereditary graph classes contained in and substitution closed. The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is given There is also offered an algebraic view on the class of all prime inductive classes of graphs of the type . [Copyright &y& Elsevier]

Abstract: The present paper deals with counting the number and orders of regular semisimple elements, together with the number of primary classes of . The approach used in this paper depends essentially on partitions of positive integers ⩽n. We give the numbers of regular semisimple elements and primary classes of for and see that the number of regular semisimple elements is an integral polynomial in q, while the number of primary classes is a rational polynomial in q. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the conjecture saying that, for any connected locally finite graph and any vertex-transitive group of automorphisms of , at least one of the following assertions holds: (1) There exists an imprimitivity system of on with finite (maybe one-element) blocks such that the stabilizer of a vertex of the factor graph in the induced group of automorphisms is finite. (2) The graph is hyperbolic (i.e., for some positive integer , the graph defined by and contains the regular tree of valency 3). Our approach to the conjecture consists in fixing a finite permutation group and considering the conjecture under the assumption that the stabilizer of a vertex of in induces on the neighborhood of the vertex a group permutation isomorphic to . In the paper we elaborate a method (the modified track method) which allows us to prove the conjecture for many groups . The paper consists of two parts. The present first part of the paper involves results on which the modified track method arguments are based, and a few first applications of the method. The second part is devoted to applications of the modified track method. [Copyright &y& Elsevier]

In this paper we study groups G generated by two subgroups A and B such that 〈 a , b 〉 is nilpotent of class at most 2 for all a ∈ A and b ∈ B . A detailed description of the structure of such groups is obtained, generalizing the classical result of Hopkins and Levi on 2-Engel groups. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the behaviour of uncountable groups of regular cardinality ℵ in which all proper subgroups of cardinality ℵ belong to a given group class X . It is proved that if every proper subgroup of G of cardinality ℵ has finite conjugacy classes, then also the conjugacy classes of G are finite, provided that G has no simple homomorphic images of cardinality ℵ. Moreover, it turns out that if G is a locally graded group of cardinality ℵ in which every proper subgroup of cardinality ℵ contains a nilpotent subgroup of finite index, then G is nilpotent-by-finite, again under the assumption that G has no simple homomorphic images of cardinality ℵ. A similar result holds also for uncountable locally graded groups whose large proper subgroups are abelian-by-finite. [ABSTRACT FROM AUTHOR]

A finite k -net of order n is an incidence structure consisting of k ≥ 3 pairwise disjoint classes of lines, each of size n , such that every point incident with two lines from distinct classes is incident with exactly one line from each of the k classes. Deleting a line class from a k -net, with k ≥ 4 , gives a derived ( k − 1 )-net of the same order. Finite k -nets embedded in a projective plane PG ( 2 , K ) coordinatized by a field K of characteristic 0 only exist for k = 3 , 4 , see Korchmáros et al. (2014). In this paper, we investigate 3-nets embedded in PG ( 2 , K ) whose line classes are in perspective position with an axis r , that is, every point on the line r incident with a line of the net is incident with exactly one line from each class. The problem of determining all such 3-nets remains open whereas we obtain a complete classification for those coordinatizable by a group. As a corollary, the (unique) 4-net of order 3 embedded in PG ( 2 , K ) turns out to be the only 4-net embedded in PG ( 2 , K ) with a derived 3-net which can be coordinatized by a group. Our results hold true in positive characteristic under the hypothesis that the order of the k -net considered is smaller than the characteristic of K . [ABSTRACT FROM AUTHOR]

Let π be a set of primes. By H. Wielandt's definition, Sylow π-theorem holds for a finite group G if all maximal π -subgroups of G are conjugate. In the paper, the following statement is proven. Assume that π is a union of disjoint subsets σ and τ and a finite group G possesses a π -Hall subgroup which is a direct product of a σ -subgroup and a τ -subgroup. Furthermore, assume that both the Sylow σ -theorem and τ -theorem hold for G . Then the Sylow π -theorem holds for G . This result confirms a conjecture posed by H. Wielandt in 1959. [ABSTRACT FROM AUTHOR]

In this paper, we study completely faithful torsion Z p 〚 G 〛 -modules with applications to the study of Selmer groups. Namely, if G is a nonabelian group belonging to certain classes of polycyclic pro- p group, we establish the abundance of faithful torsion Z p 〚 G 〛 -modules, i.e., non-trivial torsion modules whose global annihilator ideal is zero. We then show that such Z p 〚 G 〛 -modules occur naturally in arithmetic, namely in the form of Selmer groups of elliptic curves and Selmer groups of Hida deformations. It is interesting to note that faithful Selmer groups of Hida deformations do not seem to have appeared in literature before. We will also show that faithful Selmer groups have various arithmetic properties. Namely, we show that faithfulness is an isogeny invariant, and we will prove “control theorem” results on the faithfulness of Selmer groups over a general strongly admissible p -adic Lie extension. [ABSTRACT FROM AUTHOR]

A subset S of a group G invariably generates G if G = 〈 s g ( s ) | s ∈ S 〉 for each choice of g ( s ) ∈ G , s ∈ S . In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a finitely generated linear group is invariably generated by some finite set of elements if and only if it is virtually solvable. We also show that the profinite completion of an arithmetic group having the congruence subgroup property is invariably generated by a finite set of elements. [ABSTRACT FROM AUTHOR]

Abstract: Let p be an odd prime and F be a saturated fusion system over a finite p-group S with derived subgroup of prime order, excepting the case when S ≅ P × A where P is a minimal nonabelian p-group with P ∩ Ω1(P) = 1, Ω1(P) is cyclic, and A is a finite abelian p-group. In this paper, we prove that S≤F. That is, S is resistant. This generalizes a result of Stancu in the odd prime case. [Copyright &y& Elsevier]

Abstract: Let Φ be an irreducible root system of exceptional type. In this paper we calculate the set of exponents of torsion subgroups of quotients , where is the root lattice and L runs through the sublattices thereof which are generated by root differences. The maximum such exponent is of interest because it appears in the hypotheses of a result of Liebeck and Seitz concerning lifting of embeddings from finite groups of Lie type to algebraic groups. [Copyright &y& Elsevier]

Abstract: In this paper, we first characterize the subsets of the Bruhat–Tits tree of , K a complete valued field, that are the sets of fixed points of a subgroup G of . When G is irreducible, the are the “strips” in the tree. We then evaluate the form of the strip , in particular in terms of algebraic invariants of the group G. We give two applications to representation theory, one about a new generalization of Ribet's lemma on extensions, the other about the trace-convergence of a sequence of representations. [Copyright &y& Elsevier]

Abstract: The motivating question of this paper is to ask when, given a group and generating set for , one can find a Cayley map with representativity at least two (also called a strong embedding of the graph; note that in such an embedding, each face is bounded by a cycle of the graph). We first investigate some general consequences of a Cayley map’s having representativity one, which means that there exist faces that are self-adjacent (and consequently these faces are not bounded by proper cycles of the graph). We find that there exist Cayley graphs of Abelian groups all of whose Cayley maps have representativity one. This indicates that if these graphs have strong embeddings (orientable or non-orientable), then these embeddings cannot be obtained by applying the symmetry of Cayley graphs in the most natural way. This also indicates that the Cycle Double Cover of these graphs, if it exists, cannot be constructed by applying the symmetry of these graphs in the most natural way. In addition, we investigate specific consequences in the case or 3; we provide a few classes of examples of and which may be forced to have representativity at least two; and we provide a combinatorial formalism for calculating the representativity of a Cayley map without appealing to the “lifted” embedding. [Copyright &y& Elsevier]

Abstract: Let be a regular Cayley map, and let be the corresponding skew-morphism. If the power function of assumes exactly values in , then is called of skew-type . Note that regular balanced Cayley maps are of skew-type 1, and regular -balanced (including anti-balanced) Cayley maps are of skew-type 2. Those two classes of maps have been studied by many researchers. In this paper, some results on regular Cayley maps of skew-type 3 for abelian groups are given. As applications, many new regular Cayley maps for abelian groups are constructed. [Copyright &y& Elsevier]

Abstract: In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M; (RFSG) residual finiteness of Schützenberger groups of M; and (RFRL) residual finiteness of the natural actions of M on its Green's - and -classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every -class of M has finitely many - and -classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table. [Copyright &y& Elsevier]

Abstract: Let be a Fitting class and a formation. We call a subgroup of a finite group G the – -norm of G if is the intersection of the normalizers of the products of the -residuals of all subgroups of G and the -radical of G. Let π denote a set of primes and let denote the class of all finite π-groups. We call the subgroup of G the -norm of G. A normal subgroup N of G is called -hypercentral in G if either or and every G-chief factor below N of order divisible by at least one prime in π is -central in G. Let denote the -hypercentre of G, that is, the product of all -hypercentral normal subgroups of G. In this paper, we study the properties of the – -norm, especially of the -norm of a finite group G. In particular, we investigate the relationship between the -norm and the -hypercentre of G. [Copyright &y& Elsevier]

Abstract: The affine Weyl group can be realized as the fixed point set of the affine Weyl group under a certain group automorphism α with . Let be the length function of . The main results of the paper are to prove the left-connectedness of any left cell of the weighted Coxeter group in the set for any nice , to prove all the partitions with being nice and to describe all the cells of in the set . [Copyright &y& Elsevier]

Abstract: The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite. [Copyright &y& Elsevier]

Abstract: In this paper we give an extension of the Glauberman correspondence to certain characters of blocks with normal defect groups. [Copyright &y& Elsevier]

Abstract: A subgroup A of a periodic group G is said to be Sylow permutable, or S-permutable, subgroup of G if for all Sylow subgroups P of G. The aim of this paper is to establish the local nilpotency of the section for an S-permutable subgroup A of a locally finite group G. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that if and for any , or if , the Moran set on the line with packing dimension 1 are quasisymmetrically packing-minimal. [Copyright &y& Elsevier]

Abstract: Let V be a finite dimensional vector space over , H a subgroup of , λ an irreducible character of H, and the symmetry class of tensors associated with H and λ. If the vector space V is a -module, where G is a finite group, then the corresponding symmetry class of tensors naturally becomes a -module. In this paper, we study as a -module and then obtain the corresponding character. Finally we will obtain some interesting applications of the character of this -module to the character theory of finite groups and the theory of numbers. [Copyright &y& Elsevier]

Abstract: Refining the technique worked out in previous papers, we give a characterization, up to equivalence, of the factor sets σ of partial projective representation of a group G over an algebraically closed field K in terms of equalities satisfied by σ. This allows one to conclude that any component of the partial Schur multiplier is an epimorphic image of a direct power of . Moreover, it is shown that any component of is an epimorphic image of the component of the equivalence classes of the totally defined partial factor sets. Examples with cyclic G are considered, in particular, the total component is determined when G is an arbitrary finite cyclic group. In this case is a direct power of . [Copyright &y& Elsevier]

Abstract: This paper verifies the Perfect Order Subset Conjecture for Simple Groups for all but one family of finite simple groups. Specifically, for each nonabelian finite simple group G there is some N such that the cardinality of the nonempty subset of all elements of order N in G does not divide the order of G, unless G is an orthogonal group of plus type in dimension 4n, for some . [Copyright &y& Elsevier]

Abstract: For a topological group , the dual object is defined as the set of equivalence classes of irreducible unitary representations of equipped with the Fell topology. It is well known that, if is compact, is discrete. In this paper, we investigate to what extent this remains true for precompact groups, that is, dense subgroups of compact groups. We show that: (a) if is a metrizable precompact group, then is discrete; (b) if is a countable non-metrizable precompact group, then is not discrete; (c) every non-metrizable compact group contains a dense subgroup for which is not discrete. This extends to the non-Abelian case what was known for Abelian groups. We also prove that, if is a countable Abelian precompact group, then does not have Kazhdan’s property (T), although is discrete if is metrizable. [Copyright &y& Elsevier]

Abstract: The notion of -avoidance has shown up in many places in enumerative combinatorics, but the natural goal of enumerating all -avoiding posets remains open. In this paper, we enumerate graded -avoiding posets for both reasonable definitions of the word “graded.” Our proof consists of a number of structural theorems followed by some generating function computations. We also provide asymptotics for the growth rate of the number of graded -avoiding posets. [Copyright &y& Elsevier]

Abstract: The purpose of this article is to shed new light on the combinatorial structure of Kazhdan–Lusztig cells in infinite Coxeter groups W. Our main focus is the set of distinguished involutions in W, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of we assign an explicitly defined set of equivalence relations on W that altogether conjecturally determine the partition of W into left (right) cells. We are able to prove these conjectures only in a special case, but even from these partial results we can deduce some interesting corollaries. [Copyright &y& Elsevier]

Abstract: Text: Let F be a real abelian number field with Galois group G. Let be the topological closure of the image of the ring of integers of F via some fixed embedding where p is unramified in F. In Solomon (1992) [4, Conjecture 4.1], Solomon conjectured that an explicit element of , analogous to the classical Stickelberger element, annihilates where is the ideal class group of F. We prove a strengthened version of this conjecture. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=5hFhfcomnJE. [Copyright &y& Elsevier]

Abstract: Let G be a finite group and let be the set of all complex irreducible character degrees of G. Let be the set of all primes which divide some character degree of G. The prime graph attached to G is a graph whose vertex set is and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph has no triangles, then has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups. Finally, we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices. [Copyright &y& Elsevier]

Abstract: A graph is called cubic (respectively tetravalent) if all of its vertices have valency 3 (respectively valency 4). It is called vertex-transitive (respectively arc-transitive) if its automorphism group acts transitively on its vertex-set (respectively arc-set). In this paper, we combine some new theoretical results with computer calculations to determine all cubic vertex-transitive graphs of order at most 1280. In the process, we also determine all tetravalent arc-transitive graphs of order at most 640. [Copyright &y& Elsevier]

Abstract: For a cyclic group G generated by some , i.e. , the atom of a is defined as the set of all elements generating G. Given any two elements a, b of a finite cyclic group G, we study the sumset of the atom of a and the atom of b. It is known that such a sumset is a disjoint union of atoms. The goal of this paper is to offer a deeper understanding of this phenomenon, by determining which atoms make up the sum of two given atoms and by computing the exact number of representations of each element of the sumset. [Copyright &y& Elsevier]

Abstract: Motivated by the Ahrens–Szekeres Quadrangles, we shall present a variation of the -gonal family method of construction introduced by Kantor in 1980. The relation between generalized quadrangles of order and of order has been known for a long time. A geometrical description of this interrelation was given by Payne in 1971 and rests on the notion of regular points or of regular lines. In this paper we wish to develop these connections algebraically in the hope of getting more insight into them from the group-theoretical point of view. In this way we are able to characterize two classes of known -gonal configurations. [Copyright &y& Elsevier]

Abstract: A division ring D is weakly locally finite if for every finite subset S of D the division subring of D generated by S is centrally finite. We previously introduced this notion [Bui Xuan Hai, Mai Hoang Bien, Trinh Thanh Deo, On linear groups over weakly locally finite division rings, Algebra Colloq., in press] and proved that the class of weakly locally finite division rings strictly contains the class of locally finite division rings. In this paper, for a weakly locally finite division ring D, we investigate the structure of maximal subgroups of that are radical over the center of D. Our results generalize previous results pertaining to the centrally finite case. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the nontrivial solutions to a class of systems of second-order ordinary differential equations by using the critical groups and establish the existence results under the resonant case. [Copyright &y& Elsevier]

Abstract: This paper contains extensions to words on countable scattered linear orderings of two well-known results of characterization of languages of finite words. We first extend a theorem of Schützenberger establishing that the star-free sets of finite words are exactly the languages recognized by finite aperiodic semigroups. This gives an algebraic characterization of star-free sets of words over countable scattered linear orderings. Contrarily to the case of finite words, first-order definable languages are strictly included into the star-free languages when countable scattered linear orderings are considered. Second, we extend the variety theorem of Eilenberg for finite words: there is a one-to-one correspondence between varieties of languages of words on countable scattered linear orderings and pseudo-varieties of algebras. The star-free sets are an example of such a variety of languages. [Copyright &y& Elsevier]

Abstract: R. Baer and Wielandt in 1934 and 1958, respectively, considered the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this paper, for a finite group G, we define the subgroup to be the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Set . Define for . By denote the terminal term of the ascending series. It is proved that if and only if the nilpotent residual is nilpotent. Furthermore, if all elements of prime order of G are in , then G is solvable and , where is p-length of G for , where means the set of prime divisors of . [Copyright &y& Elsevier]

Abstract: A sequence A of elements an additive group G is incomplete if there exists a group element that cannot be expressed as a sum of elements from A. The study of incomplete sequences is a popular topic in combinatorial number theory. However, the structure of incomplete sequences is still far from being understood, even in basic groups. The main goal of this paper is to give a characterization of incomplete sequences in the vector space , where d is a fixed integer and p is a large prime. As an application, we give a new proof for a recent result by Gao, Ruzsa and Thangadurai on Olsonʼs constant of and partially answer their conjecture concerning . [Copyright &y& Elsevier]