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Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou [28] in even characteristic. In 2016, L. Qu [23] proposed a new approach to constructing quadratic planar functions over F 2 n . Very recently, D. Bartoli and M. Timpanella [4] characterized the condition on coefficients a , b such that the function f a , b (x) = a x 2 2 m + 1 + b x 2 m + 1 ∈ F 2 3 m [ x ] is a planar function over F 2 3 m by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, and the new approach introduced in [23] , we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over F q k , where q = 2 m with m sufficiently large (see Theorem 1.1). The first and last classes of them are over F q 2 and F q 4 respectively, while the other two classes are over F q 3 . One class over F q 3 is an extension of f a , b (x) investigated in [4] , while our proofs seem to be much simpler. In addition, although the planar binomial over F q 2 of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in [23]. [ABSTRACT FROM AUTHOR]

A classical problem, originated by Cohn's 1966 paper [1] , is to characterize the integral domains R satisfying the property: ( GE n ) “every invertible n × n matrix with entries in R is a product of elementary matrices”. Cohn called these rings generalized Euclidean, since the classical Euclidean rings do satisfy ( GE n ) for every n > 0 . Important results on algebraic number fields motivated a natural conjecture: a non-Euclidean principal ideal domain R does not satisfy ( GE n ) for some n > 0 . We verify this conjecture for two important classes of non-Euclidean principal ideal domains: (1) the coordinate rings of special algebraic curves, among them the elliptic curves having only one rational point; (2) the non-Euclidean PID's constructed by a fixed procedure, described in Anderson's 1988 paper [2] . [ABSTRACT FROM AUTHOR]

In 2006 Sommers and Tymoczko defined so called arrangements of ideal type A I stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that A I is free if the root system is of classical type or G 2 and conjectured that this is also the case for all types. This was established only very recently in a uniform manner by Abe, Barakat, Cuntz, Hoge and Terao. The set of non-zero exponents of the free arrangement A I is given by the dual of the height partition of the roots in the complement of I in the set of positive roots, generalizing the Shapiro–Steinberg–Kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated Weyl group. Our first aim in this paper is to investigate a stronger freeness property of the A I . We show that all A I are inductively free, with the possible exception of some cases in type E 8 . In the same paper from 2006, Sommers and Tymoczko define a Poincaré polynomial I ( t ) associated with each ideal I which generalizes the Poincaré polynomial W ( t ) for the underlying Weyl group W . Solomon showed that W ( t ) satisfies a product decomposition depending on the exponents of W for any Coxeter group W . Sommers and Tymoczko showed in a case by case analysis in types A n , B n and C n , and some small rank exceptional types that a similar factorization property holds for the Poincaré polynomials I ( t ) generalizing the formula of Solomon for W ( t ) . They conjectured that their multiplicative formula for I ( t ) holds in all types. In our second aim to investigate this conjecture further, the same inductive tools we develop to obtain inductive freeness of the A I are also employed. Here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions. [ABSTRACT FROM AUTHOR]

Abstract This paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors. This together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor. [ABSTRACT FROM AUTHOR]

Abstract Based on classical results of Rees and on multivariate Hilbert polynomials, we define new mixed multiplicities of two arbitrary ideals in a local ring (A , m) and use them to express the local degrees of all varieties appearing in the Gaffney–Gassler construction of Segre cycles. We prove that the classical mixed multiplicities of m and an arbitrary ideal I , which are a special case of the new ones, are equal to the generalized Samuel multiplicities of an ideal in the Rees algebra R I (A). This equality is used to improve a result of Jeffries, Montaño and Varbaro on the degree of the fiber cone of an ideal. We conclude the paper with formulas (and their inverses) which express the degrees of Segre classes of subschemes of arbitrary projective varieties by generalized Samuel multiplicities or by classical mixed multiplicities. Using the mixed multiplicities of balanced rational normal scrolls, which have been computed by Hoang and Lam, we find the mixed multiplicities of all rational normal scrolls as well as their Segre classes and their generalized Samuel multiplicities. [ABSTRACT FROM AUTHOR]

Abstract Triangular decomposition is one of the standard ways to represent the radical of a polynomial ideal. A general algorithm for computing such a decomposition was proposed by A. Szántó. In this paper, we give the first complete bounds for the degrees of the polynomials and the number of components in the output of the algorithm, providing explicit formulas for these bounds. [ABSTRACT FROM AUTHOR]

Abstract Let G ⩽ Sym (Ω) be a finite almost simple primitive permutation group with socle G 0. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G , denoted b (G) , is the minimal size of a base. We say that G is standard if G 0 = A n and Ω is an orbit of subsets or partitions of { 1 , ... , n } , or if G 0 is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G 0. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b (G) ⩽ 7 for every non-standard group G , with equality if and only if G is the Mathieu group M 24 in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b (G) = 6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type. [ABSTRACT FROM AUTHOR]

Let G be a finite group and σ = { σ i | i ∈ I } be a partition of the set of all primes P . A chief factor H / K of G is said to be σ-central (in G ) if the semidirect product ( H / K ) ⋊ ( G / C G ( H / K ) ) is a σ i -group for some i ∈ I . The group G is said to be σ-nilpotent if either G = 1 or every chief factor of G is σ -central. Let G N σ be the σ-nilpotent residual of G , that is, the intersection of all normal subgroups N of G with σ -nilpotent quotient G / N . Then we say that G is σ-supersoluble if each chief factor of G below G N σ is cyclic. In this paper we study properties of σ -supersoluble groups and also consider some applications of such groups in the theory of generalized CLT -groups. [ABSTRACT FROM AUTHOR]

A cycle base of a permutation group is defined to be a maximal set of its pairwise non-conjugate regular cyclic subgroups. It is proved in this paper that a cycle base of a permutation group of degree n can be constructed in polynomial time in n . [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]

In this paper we consider L ∞ -algebras equipped with complete descending filtrations. We prove that, under some mild conditions, an L ∞ quasi-isomorphism U : L → L ˜ induces a weak equivalence between the Deligne–Getzler–Hinich (DGH) ∞-groupoids corresponding to L and L ˜ , respectively. This paper may be considered as a modest addition to foundational paper [10] by Ezra Getzler. [ABSTRACT FROM AUTHOR]

In this paper we show that not all affine rational complex surfaces can be parametrized birational and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the curve at infinity of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct new families of irreducible Gelfand–Tsetlin modules for U q ( gl n ) . These modules have arbitrary singularity and Gelfand–Tsetlin multiplicities bounded by 2. Most previously known irreducible modules had all Gelfand–Tsetlin multiplicities bounded by 1 [16] , [17] . In particular, our method provides new families of irreducible Gelfand–Tsetlin modules for gl n . This generalizes the results of [10] and [15] . [ABSTRACT FROM AUTHOR]

In this paper, we study numerical classes of integral curves on a ruled surface P ( E ) associated to a stable bundle E . The nef cone of a ruled surface is generated by the classes of a minimal section C 0 and a fiber f . We compute the smallest integer b such that the class k C 0 + b f contains a multisecant curve for k = 2 and 3. Also we find its consequence on the Lange stability of S y m k E . [ABSTRACT FROM AUTHOR]

In this paper, we use the properties of subgroups with given order to study the structure of finite groups. The main result is as follows: Let G be a group and P be a Sylow p -subgroup of G . Suppose that 1 < d ≤ | P | . If every subgroup H of P with | H | = d is M -supplemented in G , then every non-abelian pd - G -chief factor A / B satisfies one of the following conditions: (1) A / B ≅ P S L ( 2 , 7 ) and p = 7 ; A / B ≅ P S L ( 2 , 11 ) and p = 11 ; (2) A / B ≅ P S L ( 2 , 2 t ) and p = 2 t + 1 > 3 is a Fermat prime; (3) A / B ≅ P S L ( n , q ) , n ≥ 3 is a prime, ( n , q − 1 ) = 1 and p = q n − 1 / q − 1 ; (4) A / B ≅ M 11 and p = 11 ; A / B ≅ M 23 and p = 23 ; (5) A / B ≅ A p and p ≥ 5 . [ABSTRACT FROM AUTHOR]

In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein and almost Gorenstein and to be non-level and almost Gorenstein in terms of the structure of the partially ordered set defining the Hibi ring. We also state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein in terms of the shape of the ladder. [ABSTRACT FROM AUTHOR]

Let Λ be a concealed canonical algebra and d the dimension vector of a Λ-module which is periodic respect to the action of the Auslander–Reiten translation τ . In the paper, we investigate the union of the closures of the orbits of the τ -periodic Λ-modules of dimension vector d . We show that this set is closed and regular in codimension one. [ABSTRACT FROM AUTHOR]

In this paper, we extend the theory of special local Moufang sets. We construct a local Moufang set from every local Jordan pair, and we show that every local Moufang set satisfying certain (natural) conditions gives rise to a local Jordan pair. We also explore the connections between these two constructions. [ABSTRACT FROM AUTHOR]

A right R -module is called R -projective if it is projective relative to the right R -module R R . A ring R is called right testing for projectivity if every right R -projective module is projective. Every right perfect ring is right testing and there are examples of local rings that are not right testing. In this paper an attempt is made to understand the class of right testing rings and several examples are provided to highlight the set theoretic obstacles in characterizing such rings. [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: A ring R is left exact if, for every finitely generated left submodule , every left R-linear function from S to R extends to a left R-linear function from to R. The class of exact rings generalizes that of self-injective rings and has been introduced in a recent paper by Wilding, Johnson, and Kambites. In our paper we show that the group ring of a group G over a ring R is left exact if and only if R is left exact and G is locally finite. [Copyright &y& Elsevier]

Abstract: For a set of primes π, a group X is said to be π-decomposable if is the direct product of a π-subgroup and a -subgroup , where is the complementary of π in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: “Let π be a set of odd primes. If the finite group is a product of two π-decomposable subgroups and , then and this is a Hall π-subgroup of G.” We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper. [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]

Let K be an algebraically closed field of characteristic p ⩾ 0 and let Y = Spin 2 n + 1 ( K ) ( n ⩾ 3 ) be a simply connected simple algebraic group of type B n over K . Also let X be the subgroup of type D n , embedded in Y in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for Y . In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational KY -modules on which X acts with exactly two composition factors, completing the work of Ford in [12] . [ABSTRACT FROM AUTHOR]

Let H be an atomic monoid. The set of distances Δ ( H ) of H is the set of all d ∈ N with the following property: there are irreducible elements u 1 , … , u k , v 1 … , v k + d such that u 1 ⋅ … ⋅ u k = v 1 ⋅ … ⋅ v k + d but u 1 ⋅ … ⋅ u k cannot be written as a product of ℓ irreducible elements for any ℓ ∈ N with k < ℓ < k + d . It is well-known (and easy to show) that, if Δ ( H ) is nonempty, then min ⁡ Δ ( H ) = gcd ⁡ Δ ( H ) . In this paper we show conversely that for every finite nonempty set Δ ⊂ N with min ⁡ Δ = gcd ⁡ Δ there is a finitely generated Krull monoid H such that Δ ( H ) = Δ . [ABSTRACT FROM AUTHOR]

We investigate Lie bialgebra structures on simple Lie algebras of non-split type A. It turns out that there are several classes of such Lie bialgebra structures, and it is possible to classify some of them. The classification is obtained using Belavin–Drinfeld cohomology sets, which are introduced in the paper. Our description is particularly detailed over fields of cohomological dimension at most two, and is related to quaternion algebras and the Brauer group. We then extend the results to certain rational function fields over real closed fields via Pfister's theory of quadratic forms and his solution to Hilbert's Seventeenth Problem. [ABSTRACT FROM AUTHOR]

In order to find a tractable theory to substitute for the wild character theory of the group of n × n unipotent upper-triangular matrices over a finite field F q , André and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set S of normal subgroups of G . We call such supercharacter theory the normal supercharacter theory generated by S . It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of G are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, we argue that normal supercharacter theories cannot be obtained by previously known supercharacter theory constructions. [ABSTRACT FROM AUTHOR]

In this paper, we give the fundamental normal homomorphism theorem of ordered semihypergroups. Moreover, we characterize the (strongly) ordered regular relations on an ordered semihypergroup by means of the ρ -chain, and obtain that the set of all strongly ordered regular relations on an ordered semihypergroup consists a complete lattice. Finally, the subset of an ordered semihypergroup can serve as some ordered regular (ordered semilattice) relation class is determined. [ABSTRACT FROM AUTHOR]

It is conjectured by Assem, Schiffler and Shramchenko in [3] that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of Dynkin type or rank 2, see [3] . The aim of this paper is to prove it for algebras of type A ˜ . We use triangulations of annuli and algebraic independence of clusters to prove unistructurality for algebras arising from annuli, which are of type A ˜ . We also prove the automorphism conjecture from [3] for algebras of type A ˜ as a direct consequence. [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 D be an integral domain with quotient field K. The b-operation that associates to each nonzero D-submodule E of K, , is a semistar operation that plays an important role in many questions of ring theory (e.g., if I is a nonzero ideal in D, coincides with its integral closure). In a first part of the paper, we study the integral domains that are b-Noetherian (i.e., such that, for each nonzero ideal I of D, for some a finitely generated ideal J of D). For instance, we prove that a b-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily b-Noetherian. We also characterize several distinguished classes of b-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type canonically associated to a given semistar operation ⋆ (for instance, the b-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations d, t, w and v equal to some of the canonically associated e.a.b. operations b, , and . [Copyright &y& Elsevier]

Abstract: This paper studies semigroup algebras A over a domain R for a certain family of semigroups which includes the symmetric groups, the full-transformation semigroups, and the rook semigroups. For each algebra A an A-module M on which A acts is introduced and an algebra B is defined as the commuting algebra of A acting on M. It is shown that A is also the commuting algebra of B acting on M. When A is a symmetric group algebra , B is the corresponding Schur algebra . The representation theory of the semigroup algebras A is well known. This paper analyzes the irreducible representations of the B algebras. In many cases, including the symmetric group, full-transformation semigroup, and rook semigroup cases, a complete set of inequivalent irreducible representations of the B algebra is obtained. In particular, this gives what appears to be a new description of a complete set of irreducible representations for the Schur algebra . [Copyright &y& Elsevier]

Abstract: In this paper we find the number of classes of conjugate π-Hall subgroups in all finite almost simple groups. We also complete the classification of π-Hall subgroups in finite simple groups and correct some mistakes from our previous paper. [Copyright &y& Elsevier]

Abstract: Let S and be naturally valenced association schemes on sets X and , respectively, and let ϕ be a (combinatorial) morphism from to . In Xu (2009) , a necessary and sufficient condition was given for ϕ to induce an algebra homomorphism from the scheme ring to the scheme ring . The present paper provides new techniques with which this result can be proved without assuming to be finite. To do this, we will first need to prove that for any normal closed subset T of S, whether T is finite or infinite, the quotient is a naturally valenced association scheme on the set . We will also need to discuss scheme ring homomorphisms of naturally valenced association schemes, and prove some isomorphism theorems without assuming the kernels of the scheme ring homomorphisms to be finite. As a direct consequence, for a naturally valenced commutative association scheme S on a set X and any closed subset T of S, the quotient is a naturally valenced commutative association scheme on the set . The approach in this paper is different from Xu (2009) , and quasi-algebraic morphisms of naturally valenced association schemes are also studied. [Copyright &y& Elsevier]

Abstract: This is the third in a series of four papers on fixed point ratios for non-subspace actions of finite classical groups. Our main result states that if G is a finite almost simple classical group and Ω is a faithful transitive non-subspace G-set then either for all elements of prime order, or is one of a small number of known exceptions. In this paper we consider the case where is contained in one of the Aschbacher families or . [Copyright &y& Elsevier]

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498–504] to the class of rings that are images of Cohen–Macaulay local rings. Namely, let R be a local Noetherian domain of positive characteristic that is an image of a Cohen–Macaulay local ring. We prove that all local cohomology of R (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of R is a big Cohen–Macaulay algebra. Since every excellent local ring is an image of a Cohen–Macaulay local ring, this result is a generalization of the main result of M. Hochster and Huneke in [Ann. of Math. 135 (1992), 45–79] with a simpler proof. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an even rootless lattice L is isomorphic to 2 A 1 , 2 D 4 , 2 E 8 , or BW 16 if the lattice type vertex operator algebra V L + is of class S 4 . In addition, we prove that the vertex operator algebra V 2 D 4 + is of class S 5 , and the vertex operator algebras V 2 A 1 + , V 2 E 8 + , and V BW 16 + are of class S 7 . [ABSTRACT FROM AUTHOR]

In this paper, we study simplicial complexes whose Stanley–Reisner rings are almost Gorenstein and have a -invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen–Macaulay simplicial complexes. A d -dimensional simplicial complex Δ is said to be uniformly Cohen–Macaulay if it is Cohen–Macaulay and, for any facet F of Δ, the simplicial complex Δ ∖ { F } is Cohen–Macaulay of dimension d . We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen–Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension ≤2. [ABSTRACT FROM AUTHOR]

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 Lie algebra L is said to be of breadth k if the maximal dimension of the images of left multiplication by elements of the algebra is k . In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth less than or equal to two. Furthermore, using these characterizations we determined the isomorphism classes of these algebras. [ABSTRACT FROM AUTHOR]

A formation F of finite groups is called a lattice formation if the set of all F -subnormal subgroups is a sublattice of the lattice of all subgroups in every finite group. The main result of this paper describes the family of all subgroup-closed lattice formations, and it can be regarded as an important step towards the solution of Shemetkov's classification problem. [ABSTRACT FROM AUTHOR]

In this paper we study the colimit N 2 ( G ) of abelian subgroups of a discrete group G . This group is the fundamental group of a subspace B ( 2 , G ) of the classifying space BG as introduced in [1] . We describe N 2 ( G ) for certain groups, and apply our results to study the homotopy type of the space B ( 2 , G ) . We give a list of classes of groups for which B ( 2 , G ) is not an Eilenberg–Maclane space of type K ( π , 1 ) . [ABSTRACT FROM AUTHOR]

In this paper we study graded Betti numbers of any nondegenerate 3-regular algebraic set X in a projective space P n . More concretely, via Generic initial ideals (Gins) method we mainly consider ‘tailing’ Betti numbers, whose homological index is at least codim ( X , P n ) . For this purpose, we first introduce a key definition ‘ ND ( 1 ) property’, which provides a suitable ground where one can generalize the concepts such as ‘being nondegenerate’ or ‘of minimal degree’ from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing Betti numbers. Next, we recall basic notions and facts on Gins theory and we analyze the generation structure of the reverse lexicographic (rlex) Gins of 3-regular ND ( 1 ) subschemes. As a result, we present exact formulae for these tailing Betti numbers, which connect them with linear normality of general linear sections of X ∩ Λ with a linear subspace Λ of dimension at least codim ( X , P n ) . Finally, we consider some applications and related examples. [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]

The present paper shows that if q ∈ P or q = 0 , where P is the set of prime numbers, then there exist characteristic q fields E q , k : k ∈ N , of Brauer dimension Brd ( E q , k ) = k and infinite absolute Brauer p -dimensions abrd p ( E q , k ) , for all p ∈ P not dividing q 2 − q . This ensures that Brd p ( F q , k ) = ∞ , p † q 2 − q , for every finitely-generated transcendental extension F q , k / E q , k . We also prove that each sequence a p , b p , p ∈ P , satisfying the conditions a 2 = b 2 and 0 ≤ b p ≤ a p ≤ ∞ , equals the sequence abrd p ( E ) , Brd p ( E ) , p ∈ P , for a field E of characteristic zero. [ABSTRACT FROM AUTHOR]

Let F be a field and let F 〈 X 〉 be the free unital associative F -algebra on the free generating set X = { x 1 , x 2 , … } . A subalgebra (a vector subspace) V in F 〈 X 〉 is called a T-subalgebra (a T-subspace ) if ϕ ( V ) ⊆ V for all endomorphisms ϕ of F 〈 X 〉 . For an algebra G , its central polynomials form a T -subalgebra C ( G ) in F 〈 X 〉 . Over a field of characteristic p > 2 there are algebras G whose algebras of all central polynomials C ( G ) are not finitely generated as T-subspaces in F 〈 X 〉 . However, no example of an algebra G such that C ( G ) is not finitely generated as a T-subalgebra is known yet. In the present paper we construct the first example of a 2-graded unital associative algebra B over a field of characteristic p > 2 whose algebra C 2 ( B ) of all 2-graded central polynomials is not finitely generated as a T 2 -subalgebra in the free 2-graded unital associative F -algebra F 〈 Y , Z 〉 . We hope that our example will help to construct an algebra G whose algebra C ( G ) of (ordinary) central polynomials is not finitely generated as a T -subalgebra in F 〈 X 〉 . [ABSTRACT FROM AUTHOR]

Given a finite group G and an integer e ≥ 1 dividing the order of G , the size of the set L e ( G ) = { x ∈ G | x e = 1 } was studied originally by Frobenius, in order to find restrictions on the structure of G . The aim of the present paper is to classify groups by B ( G ) = max ⁡ { | L e ( G ) | e | e ∈ Div ( exp ⁡ ( G ) ) } , where Div ( exp ⁡ ( G ) ) is the set of all divisors of the exponent exp ⁡ ( G ) of G . We will show general statements regarding the center and the central quotient of G , by looking at B ( G ) . This improves some recent contributions of W. Meng and J. Shi in [7] . [ABSTRACT FROM AUTHOR]