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Suppose that π:Y → Xis a finite map of normal varieties over a perfect field of characteristicp > 0. Previous work of the authors gave a criterion for when Frobenius splittings onX(or more generally anyp−e-linear map) extend toY. In this paper we give an alternate and highly explicit proof of this criterion (checking term by term) when π is tamely ramified in codimension 1. Some additional examples are also explored. [ABSTRACT FROM AUTHOR]

Many facts about group theory can be generalized to the context of the theory of association schemes. In particular, association schemes with fewer than 6 elements are all commutative. While there is a nonabelian group with 6 elements which is unique up to isomorphism, there are infinitely many isomorphism classes of non-commutative association schemes with 6 elements. All examples previously known to us are imprimitive, and fall into three classes which are reasonably well understood. In this paper, we construct a fourth class of noncommutative, imprimitive association schemes of rank 6. [ABSTRACT FROM AUTHOR]

In this paper we consider the transfer of the property of being a left Goldie ring between a ringAand its partial crossed productA*αGby a twisted partial action α of a groupGonA. [ABSTRACT FROM AUTHOR]

The modified quantized enveloping algebrahas a remarkable canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis offor typeA2are determined. [ABSTRACT FROM AUTHOR]

In this paper we make an attempt to study right loops (S,o) in which, for eachy ∈ S, the map σyfrom the inner mapping groupGSof (S,o) to itself given by σy(h)(x)o h(y) = h(xoy),x ∈ S,h ∈ GSis a homomorphism. The concept of twisted automorphisms of a right loop and also the concept of twisted right gyrogroup appears naturally and it turns out that the study is almost equivalent to the study of twisted automorphisms and a twisted right gyrogroup. We also study relationship between twisted gyrotransversals and twisted subgroups. [ABSTRACT FROM PUBLISHER]

One aim of this paper is to prove that for any hereditary radical property 𝒫, there exists a radical property 𝒫′ supplementing 𝒫. Certain hereditary upper radical properties of groups are constructed, and some of them are applied to formulate the necessary and sufficient condition for a hereditary radical 𝒫 and its supplementing radical 𝒫′ forming a dual pair. [ABSTRACT FROM PUBLISHER]

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined. [ABSTRACT FROM PUBLISHER]

A subgroupAof a finite groupGis called aTI-subgroup if eitherA∩Ax = 1 orA∩Ax = Aholds for allx ∈ G. In this paper, finite group all of whose meta-cyclic subgroups areTI-subgroups are classified completely. In particular, such groups are solvable. [ABSTRACT FROM PUBLISHER]

IfGis a finite group with subgroupH, then theChermak–Delgado measure of H (in G)is defined as |H||CG(H)|. The Chermak–Delgado lattice ofG, denoted 𝒞𝒟(G), is the set of all subgroups with maximal Chermak–Delgado measure; this set is a moduar sublattice within the subgroup lattice ofG. In this paper we provide an example of ap-groupP, for any primep, where 𝒞𝒟(P) is lattice isomorphic to 2 copies of ℳ2(a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in ap-group with Chermak–Delgado lattice that, for any positive integersnandl, is a 2l-string ofn-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for ap-group with Chermak–Delgado lattice that is a 2l-string of ℳp+1(quasiantichains, each of widthp + 1) adjoined maximum-to-minimum. [ABSTRACT FROM AUTHOR]

Let ℋ = ℋℂ, −1(𝔖n) be the Hecke algebra of the symmetric group 𝔖n. For partitions λ and ν with ν 2 − regular, define the Specht moduleS(λ) and the irreducible moduleD(ν). Definedλν = [S(λ):D(ν)] to be the composition multiplicity ofD(ν) inS(λ). In this paper we compute the decomposition numbersdλνfor all partitions of the form λ = (a,c, 1b) and ν 2 − regular. [ABSTRACT FROM AUTHOR]

We call the action of an automorphism α of a finite groupGa Hughes type action if it is described by conditions on the orders of elements ofG ⟨ α ⟩ − G. In the present paper we study the structure of finite groupGadmitting an automorphism α of prime orderpso that the orders of elements inG ⟨ α ⟩ − Gare not divisible byp2. [ABSTRACT FROM AUTHOR]

For a finite groupG, letv(G) denote the number of conjugacy classes of non-normal subgroups ofGandvc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups ofG. In this paper, we show that every finite groupGsatisfyingv(G) ≤2|π(G)| orvc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable groupG,v(G) = 2|π(G)| +1 if and only ifG ≅ A5. [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of ternaryHv-groups and some their properties. We give some examples of ternaryHv-groups. Also, we consider the fundamental relation β* on a ternaryHv-group and prove that β* is a compatible relation on a ternaryHv-group. In addition, we define theP-hyperoperation, and then, we construct a new ternaryHv-group. [ABSTRACT FROM AUTHOR]

The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the Weyl arrangement and their parallel translations. It was introduced by J.-Y. Shi in the study of the Kazhdan–Lusztig representation of the affine Weyl groups. M. Yoshinaga showed that the cone over every Shi arrangement is free. In this paper, we construct an explicit basis for the derivation module of the cone over the Shi arrangements of the typeBℓorCℓ. [ABSTRACT FROM AUTHOR]

The affine (Hopf) algebra of the special linear groupSL(2ℓ,k) for ℓ ≥1 is not a free module over one of its Hopf subalgebras. In this paper we give a very simple proof of this fact over a fieldkof characteristic zero. The proof boils down to showing that a certain module is not cyclic. [ABSTRACT FROM AUTHOR]

LetBbe a torsion-free Abelian group of finite rank withOT(B) = type(ℚ). This paper gives necessary and sufficient conditions for such a groupBto be a finitely faithfulS-groups with the property that every finite rank pure subgroup of a groupAwithExt(A,B) torsion-free isB-cobalanced inA. Several applications of this result are given. In particular, we classify the torsion-free groupsAof finite rank such that Ext(A,B) is torsion-free in the case thatBis an irreducible Murley group. [ABSTRACT FROM AUTHOR]

The motivation for this paper is the study of the relation between the zero component of the maximal graded algebra of quotients and the maximal graded algebra of quotients of the zero component, both in the Lie case and when considering Martindale algebras of quotients in the associative setting. We apply our results to prove that the finitary complex Lie algebras are (graded) strongly nondegenerate and compute their maximal algebras of quotients. [ABSTRACT FROM AUTHOR]

In 1999, M. Petrich and N. R. Reilly proposed some problems in their monograph “Completely regular semigroups,” one of which is: Characterize those completely regular semigroupsSwith the property that every congruence on every 𝒟-class extends to a congruence onS. The problem is solved in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree. [ABSTRACT FROM AUTHOR]

In this paper we consider various classes of monoids of transformations of a finite chain, including those of transformations that preserve or reverse either the order or the orientation. In line with Howie and McFadden [24], we complete the study of the ranks (and of idempotent ranks, when applicable) of all their ideals. [ABSTRACT FROM AUTHOR]

In this paper we present a necessary condition for ap-groupV ≤ Gto be a vertex of some indecomposable direct summand of the permutation modulekH ↑G, whereH ≤ G, andGis a finite group. We call this conditionH-suitability and present a method how to check for it. In an example, we determine allH-suitable groups. In fact, in this example everyH-suitable group is the vertex of some indecomposable direct summand ofkH ↑G. [ABSTRACT FROM AUTHOR]

LetPnandTnbe the partial transformation and the full transformation semigroups on the set {1,…,n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of heightrin the subsemigroupPK(n,r) = {α ∈Pn: |im (α)| ≤r} ofPnto be a (minimal) generating set ofPK(n,r); and similarly, for any set of full transformations of heightrin the subsemigroupK(n,r) = {α ∈Tn: |im (α)| ≤r} ofTnto be a (minimal) generating set ofK(n,r) for 2 ≤ r ≤ n − 1. [ABSTRACT FROM AUTHOR]

In this paper, we characterize finite groupsGsatisfying that, for every prime power divisorpnof the order ofG, there exists a subgroupHof indexpn(of orderpn) inGsuch thatHis normal or abnormal inG. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the definition of generalized H-Hom–Lie algebras (i.e., Hom-Lie algebras in the Yetter–Drinfeld category) for any Hopf algebra H, and obtain generalized H-Hom-Lie algebras from H-Hom-associative algebras (i.e., Hom-associative algebras in) and generalizedH-Lie algebras (i.e., Lie algebras in), respectively. We show that if A is a sum of two H-commutative Hom-associative subalgebras, then the H-commutator Hom-ideal of A is nilpotent. Finally, we describe the H-Hom-Lie ideal structures of the H-Hom-associative algebras by analogy with that of H-algebras. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to connect the notion of Gorenstein dimension with AS-Gorenstein algebras. In particular, we show that a noetherian connected graded algebra having a balanced dualizing complex is AS-Gorenstein if the balanced dualizing complex has finite Gorenstein dimension. As a preparation, we generalize the Auslander–Bridger formula to the class of noncommutative noetherian connected graded algebras having balanced dualizing complexes. [ABSTRACT FROM AUTHOR]

Letw(x,y) be a word in two variables and 𝔚 the variety determined byw. In this paper we raise the following question: if for every pair of elementsa,bin a groupGthere existsg ∈ Gsuch thatw(ag,b) = 1, under what conditions does the groupGbelong to 𝔚? In particular, we consider then-Engel wordw(x,y) = [x,ny]. We show that in this case the property is satisfied when the groupGis metabelian. Ifn = 2, then we extend this result to the class of all solvable groups. [ABSTRACT FROM AUTHOR]

LetKbe an associative and commutative ring with 1,ka subring ofKsuch that 1 ∈ k,Kis an integral finitely generated extension ofk, the element 2 invertible ink, andkis semilocal. The paper studies subgroups of the general linear groupGLn(K) withn ≥ 2 containing the special linear groupSLn(k). [ABSTRACT FROM AUTHOR]

Among other results of this paper, we single out the following ones. We show that division rings are the only division semirings over which the categories of semimodules are Schreier varieties, i.e., all subsemimodules of free semimodules are free too. We a complete description of division semiringsRover which the categories of semimodules ℳRarep-Schreier varieties, i.e., varieties whose all projective algebras are free. We give a complete description of proper division semiringsRwhose categories of semimodules ℳR(X)over the polynomial semiringsR(X) overR, in not necessary commuting variablesX, arep-Schreier varieties. We show that the categories of semimodules ℳR(X)over the polynomial semiringsR(X) overN-valued semiringsR, in particular ℳN(X), arep-Schreier varieties. We also show that forN-valued semiringsS, the semimodule categories ℳSnever are Schreier varieties. [ABSTRACT FROM PUBLISHER]

In this paper, we construct derived equivalences between two subrings of relevant Φ-Yoneda rings from an arbitrary short exact sequence in an abelian category. As a consequence, any short exact sequence in an abelian category gives rise to a derived equivalence between two subrings of endomorphism rings. [ABSTRACT FROM PUBLISHER]

Using the Tate–Poitou duality, Sansuc proved that the group III1(K,G) is stablyK-birational invariant ofGfor a connected linear algebraic group defined over a number fieldK. In this paper, we consider the case whereKis a function field, in one variable over a PAC field (Kis a “good” field in sense Colliot–Thélène and Kunyavskii). We show that the group III1(K,G) is stablyK-birational invariant whenGis a connected reductiveK-group. Since we no longer have the Tate–Poitou duality at our disposition, we use the flasque resolutions of Colliot–Thélène and Sansuc. En utilisant la dualité de Tate–Poitou, Sansuc a établi le caractère d'invariant stablementK-birationnel du groupe III1(K,G) pour un groupe algébrique linéaire connexeGdéfini sur un corps de nombresK. Dans cet article, nous considérons le cas oùKest un corps de fonctions en une variable sur un corps PAC (Kest un “bon” corps au sens de Colliot–Thélène et Kunyavskii). Nous montrons le caractère d'invariant stablementK-birationnel du groupe III1(K,G) pour unK-groupe réductif connexeG. Comme nous n'avons plus à notre disposition la dualité de Tate–Poitou, nous devons utiliser les résolutions flasques de Colliot–Thélène et Sansuc. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is the study of hypergroups associated with hypergraphs. In this regards, we construct a ρ-hypergroup by means a given hypergraph by defining a special relation ρ, and then we investigate some related properties. Further, we introduce a special product of ρ-hypergroups. Also, we bridge between subhypergraphs and subhypergroups. Finally, the fundamental relation of a ρ-hypergroup is studied. [ABSTRACT FROM AUTHOR]

A finite oscillator dictionary which has important applications in sequences designs and the compressive sensing was introduced by Gurevich, Hadani, and Sochen in [3]. In this paper, closed formulae of the finite split oscillator dictionary 𝔖sare revisited by a simple proof, the structure of nonsplit tori of the groupSL(2, 𝔽p) is studied, and an explicit algorithm for computing the finite nonsplit oscillator dictionary 𝔖nsis described. [ABSTRACT FROM AUTHOR]

LetE/Fbe a Galois extension of number fields with the quaternion Galois groupQ8. In this paper, we prove some relations connecting orders of the odd part of the kernel of the transfer map of the tame kernel ofEwith the same orders of some of its subfields. LetE/ℚ be a Galois extension of number fields with the Galois groupQ8andpan odd prime such thatp ≡ 3 (mod 4). We prove that if there is at most one quadratic subfield such that thep-Sylow subgroup of the tame kernel is nontrivial, thenpr-rank(K2(E/K)) is even, i.e., 2|pr-rank(K2(𝒪E)) − pr-rank(K2(𝒪K)), whereKis the quartic subfield ofE. [ABSTRACT FROM PUBLISHER]

Finite groupsGsuch thatG/Z(G) ≈C2 × C2whereC2denotes a cyclic group of order 2 andZ(G) is the center ofGwere studied in [5] and were used to classify finite loops with alternative loop algebras. In this paper we extend this result to finitely generated groups such thatG/Z(G) ≈Cp × CpwhereCpdenotes a cyclic group of prime orderpand provide an explicit description of all such groups. [ABSTRACT FROM AUTHOR]

LetGbe a group. By using a family 𝒜 of subsets of automorphisms ofG, we introduced a simple graph Γ𝒜(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of Γ𝒜(G). [ABSTRACT FROM AUTHOR]

In the paper we describe the subcategory of the category of ℤ-graded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the Tits, Kantor, and Koecher TKK construction. For instance, we prove thatL = L−1⊕L0⊕L1can be constructed from a Jordan pair if and only ifL0 = [L−1,L1] and the second graded homology groupis trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras. New functorial versions of the TKK construction are given for pairs and algebras. [ABSTRACT FROM AUTHOR]

A subgroupHof a finite groupGis calledc*-normal inGif there exists a normal subgroupKofGsuch thatG = HKandH∩KisS-quasinormally embedded inG. In this paper, the structure of a finite groupGwith somec*-normal subgroups of Sylowp-subgroups with an any fixed order is characterized and some known related results onp-nilpotency of finite groups are generalized. [ABSTRACT FROM AUTHOR]

Abelian groups whose endomorphism rings are von Neumann regular have been extensively investigated in the literature. In this paper, we study modules whose endomorphism rings are von Neumann regular, which we call endoregular modules. We provide characterizations of endoregular modules and investigate their properties. Some classes of ringsRare characterized in terms of endoregularR-modules. It is shown that a direct summand of an endoregular module inherits the property, while a direct sum of endoregular modules does not. Necessary and sufficient conditions for a finite direct sum of endoregular modules to be an endoregular module are provided. As a special case, modules whose endomorphism rings are semisimple artinian are characterized. We provide a precise description of an indecomposable endoregular module over an arbitrary commutative ring. A structure theorem for extending an endoregular abelian group is also provided. [ABSTRACT FROM PUBLISHER]

In this paper we characterize the paramedial, comedial, and coparamedial pairs of quasigroup operations and paramedial, comedial, and coparamedial algebras with the quasigroup operations. Then we characterize paramedial, comedial, and coparamedial algebras with a unit element and an idempotent regular element. [ABSTRACT FROM PUBLISHER]

A power automorphism θ of a group G is said to be pre-fixed-point-free if CG(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms. [ABSTRACT FROM PUBLISHER]

The notion of cohomological periodicity after 1-step has been studied by Talelli in [7, 8], and [9]. If a group G has periodic cohomology after 1-step, then G is the fundamental group of a graph of finite groups, which have periodic cohomology of the same period. Also, the fundamental group of a tree of finite groups, which have periodic cohomology of the same period, has periodic cohomology after 1-step. In this paper, we show that if a group G has only cyclic finite subgroups and is the fundamental group of a certain tree of groups, which have -steps. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a detailed structure description of derived subpolygroups of polygroups. We introduce the concept of perfect and solvable polygroups and we give some results in this respect. Finally, we discuss on τ-multi-semi-direct hyperproduct of polygroups. [ABSTRACT FROM AUTHOR]

In [1], the authors proved that the maximum sum of element orders on finite groups of the same order occurs in cyclic group. In this paper we obtain the structure of groups having maximum sum of element orders on noncyclic nilpotent group of the same order. [ABSTRACT FROM AUTHOR]

A subgroupHofGis called ℳp-supplemented inGif there exists a subgroupBofGsuch thatG = HBandTB < Gfor every maximal subgroupTofHwith |H:T| =pα. In this paper, we investigate the influence of ℳp-supplemented subgroup and some conditions forp-nilpotency andp-supersolvability of finite groups are obtained. [ABSTRACT FROM AUTHOR]

The notion of almost left factorizability and the results on almost left factorizable weakly ample semigroups, due to Gomes and the author, are adapted for restriction semigroups. The main result of the paper is that each restriction semigroup is embeddable into an almost left factorizable restriction semigroup. This generalizes a fundamental result of the structure theory of inverse semigroups. [ABSTRACT FROM AUTHOR]

An element of a group is called an iicentral if the conjugucy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camilla group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camilla groups, which are not locally solvable. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for the existence and uniqueness of normal forms of sequences of HNN extensions defined by Bokut'. Furthermore, we show that under an assumption, which holds for various applications, such normal forms always exist (but might not be unique). The conditions are amenable to be used in automatic theorem provers. We discuss also how to obtain a Grobner-Shirshov basis from the rewrite rules of Bokut' normal forms under certain assumptions. Finally, we provide an application drawn from a paper of Aanderaa and Cohen to illustrate the sufficiency conditions. [ABSTRACT FROM AUTHOR]

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) - n, where μ(G) is the minimum possible number of generators of G. Let G be fg; then Hom(G, SL2) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer. [ABSTRACT FROM AUTHOR]

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S∖{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]]. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. An algebraic element α over R is called an “integral algebraic element” over R if there exists a minimal polynomial of α over R which is monic. Let R be a direct product of m indecomposable rings Ri, m ∈ ℕ. Denote by RG the group ring of G over R and by R* the multiplicative group of R.Let G be a finite Abelian group of exponent n and n ∈ R. In this paper we give a decomposition of RG, up to isomorphism, into a direct sum of extensions of the ring R, taking into account the number of the repetitions of these extensions. If the ring R is a field, then this result is proved in Perlis and Walker (1950).Let G be a splitting Abelian group with a torsion subgroup G0. Denote by Gp the p -component of G. We give a description of the unit group U(RG) of RG in the following cases: when Ri is a ring of prime characteristic pi, G0/Gpi is finite and the exponent of G0/Gpi belongs to Ri*;when Ri is of characteristic zero, Ri has no nilpotents, G0 is finite of exponent n and n ∈ Ri*.For the establishment of these results we prove that if the ring R is indecomposable and n ∈ R*, then: the cyclotomic polynomial Φn(x) has a unique decomposition in a product of monic irreducible factors over R; andif α and β are integral algebraic elements over R which are roots of monic irreducible divisors of the cyclotomic polynomial Φn(x), then the rings R[α] and R[β] are isomorphic. [ABSTRACT FROM AUTHOR]