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This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced to classifying the solvableChermak–Delgado simple groups, which the author defines. The notion of “Chermak–Delgado simple,” or “CD-simple” for short, is a generalization of simple groups through the Chermak–Delgado lattice. The author completes a classification of Chermak–Delgado simple groups under certain restrictions on the primes involved in the group order. [ABSTRACT FROM PUBLISHER]

In a recent paper by the authors, the associative and the Lie algebras of Weyl type A[D] = A ⊗ F[D] were introduced, where A is a commutative associative algebra with an identity element over a field F of any characteristic, and F[D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, a class of the above associative and Lie algebras A[D] with F being a field of characteristic 0 and D consisting of locally finite derivations of A, is studied. The isomorphism classes of these associative and Lie algebras are determined. The structure of these algebras is described explicitly. [ABSTRACT FROM AUTHOR]

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]

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]

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]

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations. [ABSTRACT FROM AUTHOR]

A group in which all cyclic subgroups are 2-subnormal is called a 2-Baer group. The topic of this paper are generalized 2-Baer groups, i.e., groups in which the non-2-subnormal cyclic subgroups generate a proper subgroup of the group. If this subgroup is non-trivial, the group is called a generalizedT2-group. In particular, we provide structure results for such groups, investigate their nilpotency class, and construct examples of finitep-groups which are generalizedT2-groups. [ABSTRACT FROM AUTHOR]

Let, whereSnis the symmetric group onXnandPKn,ris the semigroup of partial mapssuch that |im(α)| ≤ rfor 1 ≤ r ≤ n − 1. In this paper, we find necessary and sufficient conditions for any subset ofPKn,rto be a (minimal) relative generating set of the subsemigroupPTn,rmoduloSn. Then, for each 1 ≤ r ≤ n − 1, we show that the smallest number of elements ofPKn,rwhich, together withSn, generatePTn,riswherepr(n − s) is the number of partitions ofn − swithrterms. We also consider, whereAnis the alternating group onXn. [ABSTRACT FROM AUTHOR]

A groupGis said to be aB(n,k) group if for anyn-element subsetAofG, |A2| ≤k. In this paper, a characterization ofB(5, 18) groups is given. It is shown thatGis aB(5, 18) group if and only if one of the following statements holds: (1)Gis abelian; (2) |G| ≤18; (3)G ≅ ⟨ a,b | a5 = b4 = 1,ab = a−1 ⟩. [ABSTRACT FROM AUTHOR]

In this paper we study the new notion of a symmetric pair of objects in the Yetter–Drinfeld categoryover a weak Hopf algebraHwith a bijective antipode. Then we show that the (co)commutativity and trivial property ofHare determined by some symmetric pairs of objects in, generalizing the main results in Cohen and Westreich and Pareigis. Finally, we study theu-condition generalizing the corresponding results appeared in Cohen and Westreich. [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]

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]

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]

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 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]

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]

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]

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]

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]

Extending the notion of probability to the automorphisms of a group, we find the probability of an arbitrarily chosen automorphism of a group fixing an arbitrary element of the group. [ABSTRACT FROM PUBLISHER]

We prove that all automorphisms of the automorphism group of an infinitely generated free abelian group are inner. [ABSTRACT FROM PUBLISHER]

Although it is well known that the growth of Thompson's group F is exponential, the exact growth function is still unknown. Elements of its submonoid of positive words can be described using a binary rooted tree, whose norm can be computed assigning weights to each caret. Combining this fact with a combinatorial argument, the growth function of the submonoid is computed and thus providing a first step in the computation of the growth function of the group, as well as a lower bound for the growth rate for the group. [ABSTRACT FROM AUTHOR]

The maximality of certain symplectic subgroups of unitary groups PSUn(K), n ≥ 4, (K any field admitting a non-trivial involutory automorphism) belonging to the class c5 of Aschbacher is proved. Furthermore some related geometry in the case n = 4 and K finite is investigated. [ABSTRACT FROM AUTHOR]

Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M, is a subhypermodule of M. We introduce the concept of an essential subhypermodule in M relative to an arbitrary subhypermodule T of M, which is called a T-essential subhypermodule of M. Our main goal in this work is to investigate properties of (relative) essential subhypermodules. We apply this concept to introduce extending hypermodules. Examples are provided to illustrate different concepts. [ABSTRACT FROM AUTHOR]

IfXis a subgroup of a groupG, the cardinal numberis called the normal oscillation ofXinG. The structure of soluble groups in which all subnormal subgroups have finite normal oscillation is investigated here, taking as a model the known theory ofT-groups, i.e., groups in which normality is a transitive relation. [ABSTRACT FROM AUTHOR]

The main aim of this article is to study quantitative structure of small Ree Groups2G2(q). Here, we prove that small Ree groups are uniquely determined by their orders and the set of the number of elements of the same order. [ABSTRACT FROM AUTHOR]

For a given groupGand a monomorphismφ:G→G×Gthere is a groupℱφ(G), introduced by the author, which blends Thompson’s groupFwithG. Given a presentation ofGwe determine a presentation ofℱφ(G). In particular, we prove thatℱφ(G) is finitely generated (resp. finitely presented) ifGis finitely generated (resp. finitely presented). [ABSTRACT FROM AUTHOR]

Systems of equations and their solution sets are studied in polyadic groups. We prove that a polyadic group (G,f) = der𝜃,b(G,⋅) is equational noetherian, if and only if the ordinary group (G,⋅) is equational noetherian. The structure of coordinate polyadic group of algebraic sets in equational noetherian polyadic groups is also determined. [ABSTRACT FROM PUBLISHER]

We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra Λsobtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of Λsby generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of Λsis finitely generated as an algebra. [ABSTRACT FROM AUTHOR]

We shall determine the ring structure of the Hochschild cohomology rings of integral group rings of wreathed 2-groups. [ABSTRACT FROM AUTHOR]

In this article, we consider solvable groups that have four character degrees and derived length four. [ABSTRACT FROM AUTHOR]

A finite group G is said to he a B(n, k) group if for any n-element subset {a1,...,an} of G, |{aiaj|1≥ i, j ≥ n}| ≥ k. In this article, we give characterizations of the B(5, 19) 2-groups, and the 11(6, k) 2-groups for 21 ≥ k ≥ 28. [ABSTRACT FROM AUTHOR]

We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a nontrivial partition, and symmetric and alternating groups. Also, for such a group, the corresponding proper power graph has diameter at most 26 whenever it is connected. [ABSTRACT FROM AUTHOR]

We calculate the rank gradient andp-gradient of free products with amalgamation over an amenable subgroup and HNN extensions with an amenable associated subgroup. The notion of cost is used to compute the rank gradient of amalgamated free products and HNN extensions. For thep-gradient the Kurosh subgroup theorems for amalgamated free products and HNN extensions will be used. [ABSTRACT FROM AUTHOR]

LetAbe an abelian variety over ℂ such that the semisimple part of the Hodge group ofAis a product of copies ofSU(p, 1) for somep > 1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology ofAis isomorphic to a Hodge structure in the cohomology of some abelian variety. [ABSTRACT FROM AUTHOR]

We study the rigid property of rings in the concrete, via power series rings, matrix rings, and Insertion-of-Factors-Property. In the procedure, we concentrate our attention on the skew power-serieswise Armendariz property. We next apply the McCoy condition to skew power series rings, which induces a generalization of skew power-serieswise Armendariz. The relationship is investigated among the properties above and related ring properties, and several known results are obtained as consequences of our results. [ABSTRACT FROM AUTHOR]

A groupGis said to be ann⊗-Engel, if [y,n−1x] ⊗ x = 1 for allx,y ∈ G, and we say a groupGis tensor nilpotent of class at mostn, if. In this article, we show that ifGis a 3⊗-Engel group, then ⟨ x,xy ⟩ is tensor nilpotent of class at most 2, for allx,y ∈ G. We also prove that ifGis a 4⊗-Engel group andG ⊗ Gis torsion-free, then ⟨ x,xy ⟩ is tensor nilpotent of class at most 4, for allx,y ∈ G. [ABSTRACT FROM AUTHOR]

Cyclic posets are generalizations of cyclically ordered sets. In this article, we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ringR. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14], the infinity-gon of [12], and them-cluster category of typeA∞(m ≥ 3) [13] are examples of this construction as well as some new examples such as the cluster category of ℤ2. An extension of this construction and further examples are given in [16]. [ABSTRACT FROM AUTHOR]

The main result in this article gives a partial answer to the following question forn = 2. Letk[U] andk[V] be the coordinate rings of the irreducible algebraic setsUandV, respectively. Is projective dimension ofJn(k[U × V]) finite whenever projective dimension ofJn(k[U]) and projective dimension ofJn(k[V]) are finite? [ABSTRACT FROM AUTHOR]

In this article, we study graded Clifford algebras with a gradation preserving action of automorphisms given byHp, the Heisenberg group of orderp3withpprime. After reviewing results in dimensions 3 and 4, we will determine the graded Clifford algebras that areAS-regular algebras of global dimension 5 and generalize certain results to arbitrary dimensionp. [ABSTRACT FROM AUTHOR]

We study injective hulls of simple modules over differential operator ringsR[θ;d], providing necessary conditions under which these modules are locally Artinian. As a consequence, we characterize Ore extensions ofS = K[x][θ; σ,d] for σ aK-linear automorphism anddaK-linear σ-derivation ofK[x] such that injective hulls of simpleS-modules are locally Artinian. [ABSTRACT FROM AUTHOR]

LetSbe any semigroup anda,s ∈ S. Ifa = asa, thensis anassociateofa. A subgroupGofSis anassociate subgroupofSif everya ∈ Shas a unique associatea* inG. It turns out thatG = Hzfor some idempotentz, thezenithofS. The mappinga → a* is a unary operation onS. We say thatSismonogenicifSis generated, as a unary semigroup, by a single element. We embark upon the problem of the structure of monogenic semigroups in this sense by characterizing monogenic ones belonging to completely simple semigroups, normal cryptogroups, orthogroups, combinatorial semigroups, cryptic medial semigroups, cryptic orthodox semigroups, and orthodox monoids. In each of these cases, except one, we construct a free object. The general problem remains open. [ABSTRACT FROM AUTHOR]