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Following a suggestion in a footnote of the paper [1] by Fano, we give a direct proof of the fact that one of the two Fano threefolds of degree 72 in P 38 is isomorphic to the anticanonically embedded weighted projective space P (1 , 1 , 4 , 6) . [ABSTRACT FROM AUTHOR]

In the present paper corrected versions of the statements in the paper "Description of finite-dimensional inner Rickart and Baer Jordan algebras" by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J = F p + ̇ D , J is an inner RJ-algebra if and only if, for any nonzero a ∈ D , a 2 = 0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

AbstractIn a recent paper, Baker and Oliver Lorscheid showed that Descartes’s Rule of Signs and Newton’s Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebraic structures, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well. [ABSTRACT FROM AUTHOR]

In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : (N μ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q1 and Q2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form (α , α , γ) . [ABSTRACT FROM AUTHOR]

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. [ABSTRACT FROM AUTHOR]

In this paper, we progress in the general study of rational functional identities in the setting of division rings. The main theorem in the paper states that the only additive maps f and g satisfying that f (x) = − x n g (x − 1) , for n = 3 or 4, are the zero maps. In addition, we provide an alternative proof to the result proved by Dar and Jing regarding the classification of additive maps f and g satisfying that f (x) = − x 2 g (x − 1) . [ABSTRACT FROM AUTHOR]

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we construct a family of quasi-strongly regular Cayley graphs which is defined on a finite group G with respect to a proper subgroup H of G and denoted by ΓH(G) . We also compute the full automorphism group and characterize various transitivity properties of ΓH(G) for any group G and for any subgroup H of G. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, all biderivations of solvable Lie algebras of maximal rank g=Q⊕N are characterized. Namely, we consider a solvable Lie algebra of the form g=Q⊕N , where Q is the maximal torus subalgebra of g, N is the nilradical of g and dimQ=dimN/N2 .In the case dimQ=N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given. [ABSTRACT FROM AUTHOR]

AbstractIn the first part of the paper, we provide a fairly complete description of a 2-torsion free simple ring R with involution f such that the skew traces relative to f of all elements of R form a commutative subset. In the remaining part, the results carried out earlier are put to work to delineate the structure of a simple Artinian ring R with involution f, which is equipped with a given non central submonoid M (= multiplicatively closed subset containing 1) preserved under f and under all inner automorphisms of R, and M is such that the skew traces of all elements of M form a commutative subset. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we investigate the attached prime ideals of inverse polynomial modules over skew Ore polynomials. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we introduce and explore a category of rings termed SFP-rings, defined by the property that each S-finite ideal of them is projective. We demonstrate that all hereditary rings inherently qualify as SFP-rings, while conversely, every SFP-ring is inherently semi-hereditary. Our research delves into the application and relevance of this concept across various facets of commutative ring extensions. These include localizations, direct products, trivial ring extensions, pullbacks, and the specific case of amalgamated duplication of a ring along an ideal. A significant outcome of our study is the identification and characterization of unique families of rings: those that are non-hereditary yet qualify as SFP-rings, and those that are semi-hereditary but do not meet the criteria for being SFP-rings. This work not only extends the theoretical understanding of ring structures but also contributes to the broader field of algebraic theory with practical examples and applications. [ABSTRACT FROM AUTHOR]

AbstractThe present article is devoted to introduce the notion of Hopf bracoid in the braided monoidal framework as the quantum version of skew bracoids, which have been presented by Martin-Lyons and Truman. Taking into account that Hopf braces are particular examples of Hopf bracoids, in this paper we generalize many properties of Hopf braces to the Hopf bracoid setting, we study the relationships between Hopf bracoids and skew bracoids and, finally, we obtain a categorical isomorphism between certain full subcategories of the Hopf bracoids category and the 1-cocycles category. [ABSTRACT FROM AUTHOR]

AbstractThis paper starts by showing that, for algebras in a certain class, the concepts of weak nilpotency and nilpotency coincide. It goes on to describe some solvability and nilpotency properties of bicommutative algebras, of assosymmetric algebras and of Novikov algebras and to introduce a Frattini theory for all of them. A description is also given for semisimple bicommutative algebras over any field. [ABSTRACT FROM AUTHOR]

AbstractA subgroup X of a group G is called pronormal if X and Xg are conjugate in 〈X,Xg〉 for each element g of G. Pronormal subgroups play a relevant role in many problems of group theory; for instance, they arise naturally in the study of groups in which normality is a transitive relation, because a subgroup is normal if and only if it is subnormal and pronormal. Groups with only pronormal subgroups have been described by Kuzennyĭ and Subbotin (at least in the locally graded case) and the aim of this paper is to investigate the structure of uncountable locally graded groups of cardinality ℵ in which all subgroups of cardinality ℵ are pronormal. [ABSTRACT FROM AUTHOR]

AbstractA ring is called left fusible if each of its nonzero elements is the sum of a left zero-divisor and a non-left zero-divisor. This paper aims to extend the existing theory of left fusible rings and related notions such as being left unit-fusible, being regular left fusible, and being uniquely left fusible. New examples of these rings are presented. A new notion of left generalized unit-fusible rings is introduced and discussed. The asymmetry of these notions is addressed. [ABSTRACT FROM AUTHOR]

AbstractLet G and A be finite groups of relative coprime orders and A acts on G via automorphisms. In this paper, we investigate the p-nilpotency and solvability of finite groups in which every maximal A-invariant subgroup of a Sylow p-subgroup of G for some prime p is S-semipermutable. [ABSTRACT FROM AUTHOR]

AbstractLet k be an arbitrary field and let b>1,n>1 and a be three positive integers. In this paper we explicitly describe a minimal S-graded free resolution of the semigroup algebra k[S] when S is a generalized repunit numerical semigroup, that is, when S is the submonoid of N generated by {a1,a2,…,an} where a1=∑j=0n−1bj and ai−ai−1=a bi−2, i=2,…,n, with gcd(a,a1)=1. [ABSTRACT FROM AUTHOR]

In this paper, we define a partially capable Lie superalgebra. As an application, we classify all capable nilpotent Lie superalgebras of dimension less than or equal to five. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate some properties of relative Rota-Baxter operators on Leibniz conformal algebras with respect to representations and their connections with Leibniz dendriform conformal algebras. Next, we introduce the notion of a twisted relative Rota-Baxter operator and construct a conformal NS-Leibniz algebra structure related to twisted relative Rota-Baxter operators. Furthermore, we define the cohomology of a twisted relative Rota-Baxter operator with coefficients in a suitable representation. Finally, we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationship among three n-cluster tilting subcategories of triangulated categories in a recollement. Let (D ′ , D , D ″) be a recollement of triangulated categories. We construct an n-cluster tilting subcategory of D by gluing together n-cluster tilting subcategories of D ′ and D ″ under certain conditions. On the other hand, we prove that an n-cluster tilting subcategory of D can induce n-cluster tilting subcategories of D ′ and D ″ . Using these two results, we give a one-to-one correspondence between some n-cluster tilting subcategories of D and pairs of n-cluster tilting subcategories of D ′ and D ″ . Furthermore, we obtain a recollement of abelian categories from a recollement of triangulated categories. [ABSTRACT FROM AUTHOR]

Let D be an F-central division algebra. In this paper, we investigated the exponent of the group G (D) = D * / Nrd D (D *) D ′ , where D * is the group of units of D, Nrd D (D *) is the image of D * under the reduced norm map and D ′ is the commutator subgroup of D * . We show that if exp (G (D)) < ind (D) , then D and F satisfy strong conditions. In particular, we observe that if D is a sum cyclic algebras in Br (F) , then exp (G (D)) < ind (D) if and only if F is euclidean and D is a tensor product of an ordinary quaternion algebra and a division algebra of odd index. [ABSTRACT FROM AUTHOR]

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite. [ABSTRACT FROM AUTHOR]

Let F be a saturated fusion system over a p-group S and let E be a subsystem of F with Aut E (S) = Aut F (S) . We prove that the fusion system F is generated by all E -morphisms and all F -automorphisms of some domestic intersections (called F ∖ E -domestic intersections in the paper) of F . We also get some results about control of fusion in finite groups as applications. [ABSTRACT FROM AUTHOR]

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry. [ABSTRACT FROM AUTHOR]

In this paper, we introduce proximal groups that are a generalization of topological groups. A straight-forward application of proximal groups is given in the concise representation of collections of overlapping Hilbert envelope lobes attached to the peak points on oscillatory motion waveforms that occur in sequences of video frames that track object movements. [ABSTRACT FROM AUTHOR]

Let R be an associative unital ring, and let f ∈ R [ x ] . We say that f is a left annihilating content (AC) polynomial if f = af1 for some a ∈ R and f 1 ∈ R [ x ] with l R [ x ] (f 1) = 0 . The ring R is called a left EM-ring if each f ∈ R [ x ] is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element a ∈ R and a finitely generated right ideal J of R with l R (J) = 0 and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring R [ G ] is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, R [ x ] , R [ x , x − 1 ] , and R [ [ x ] ] are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings). [ABSTRACT FROM AUTHOR]

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. [ABSTRACT FROM AUTHOR]

This paper aims to introduce the concept of nilpotency and capability in multiplicative Lie algebras. Also, we see the existence of covers of a multiplicative Lie algebra and thoroughly examine their relationships with capable and perfect multiplicative Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group. [ABSTRACT FROM AUTHOR]

In this paper, we study the generators of the module of derivations of some one-dimensional local domains. [ABSTRACT FROM AUTHOR]

Motivated by recent work on compatible mock-Lie algebras such that any linear combination of the two mock-Lie structures is a mock-Lie structure, in this paper we introduce compatible mock-Lie bialgebras as an analogue of compatible Lie bialgebras. They can also be regarded as a "compatible version" of mock-Lie bialgebras. Constructions of compatible mock-Lie bialgebras are presented via Manin triples and matched pairs. [ABSTRACT FROM AUTHOR]

AbstractA finite group G is an exact product of two subgroups A and B if G = AB and A∩B={1G}. In this paper, for all odd numbers k≥3, using the theory of skew morphisms and associated extended power functions, we present a classification of exact products of a cyclic group and a dihedral group D2k. As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified. [ABSTRACT FROM AUTHOR]

In this paper, a kind of restricted wreath product of an arbitrary band of groups by an arbitrary generalized inverse semigroup is introduced first. This wreath product turns out to be an E-solid semigroup, and if one comes out at the beginning with a normal band of groups, then it shows itself to be also a locally inverse semigroup. Then it is proved that every E-solid locally inverse semigroup S can be embedded in such a restricted wreath product of a normal band of groups Q by a generalized inverse semigroup K. Here K can be taken to be the greatest orthodox semigroup homomorphic image of S and Q can be taken to be the preimage of the band of idempotents of K under the canonical homomorphism of S onto its quotient K. In this way, the given E-solid locally inverse semigroup S decomposes into a restricted wreath product of its components Q and K whose structure derives directly from that of S. [ABSTRACT FROM AUTHOR]

Abstract–In this paper, based on the classifications of Kn-simple groups for n≤6 and under the case that every maximal subgroup has a π-nilpotent trace, we obtained the necessary and sufficient conditions on solvable groups and proved that two unsolvable problems involving groups, rings and fields are true. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we use the Balinsky-Novikov construction to naturally realize the super Heisenberg-Virasoro algebra g . We then proceed to compute the second cohomology of g with trivial coefficients and determine its universal central extension. This yields two odd 2-cocycles in addition to the Virasoro 2-cocycle. We also calculate the first cohomology group of g in the adjoint module to classify all derivations of g . Finally, we classify all automorphisms of g and describe its automorphism group. [ABSTRACT FROM AUTHOR]

AbstractThis paper is concerned with generalizing the results for Lie CT-algebras to Leibniz algebras. In some cases our results give a generalization even for the case of a Lie algebra. Results on A-algebras are used to show every Leibniz CT-algebra over an algebraically closed field of characteristic different from 2,3 is solvable or is isomorphic to sl2(F). A characterization is then given for solvable Leibniz CT-algebras. It is also shown that the class of solvable Leibniz CT-algebras is factor closed. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we make explicit an application of the wreath product construction to the Galois groups of field extensions. More precisely, given a tower of fields F⊆K⊆L with L/F finite and separable, we explicitly construct an embedding of the Galois group Gal(Lc/F) into the regular wreath product Gal(Lc/Kc)≀rGal(Kc/F) . Here Lc (resp. Kc ) denotes the Galois closure of L/F (resp. K/F). Similarly, we also construct an explicit embedding of the Galois group Gal(Lc/F) into the smaller sized wreath product Gal(Lc/K)≀ΩGal(Kc/F) , where Ω=HomF(K,Kc) is acted on by composition of automorphisms in Gal(Kc/F) . Moreover, when L/K is a Kummer extension we prove a sharper embedding, that is, that Gal(Lc/F) embeds into the wreath product Gal(L/K)≀ΩGal(Kc/F) . As corollaries we obtain embedding theorems when L/K is cyclic and when it is quadratic with char(F)≠2 . We also provide examples of these embeddings and as an illustration of the usefulness of these embedding theorems, we survey some recent applications of these types of results in field theory, arithmetic statistics, number theory and arithmetic geometry. [ABSTRACT FROM AUTHOR]

AbstractLet (A,B,C) be a recollement of extriangulated categories. In this paper, we provide bounds on the coresolution dimensions of the subcategories involved in A, B , and C . We show that a hereditary cotorsion pair in B can induce hereditary cotorsion pairs in A and C under certain conditions. Besides, we construct recollements of abelian categories from a recollement of extriangulated categories with respect to the hearts of cotorsion pairs. [ABSTRACT FROM AUTHOR]

AbstractLet M be a right R-module. We call a summand A of M a special summand if there exists a summand B of M with A≅B and A∩B=0. An internal direct sum ⊕i∈IAi of submodules Ai of M is called a special local summand if each Ai is a special summand and ⊕i∈FAi is a summand of M for any finite subset F⊆I. In this paper we give two new classes of modules for which the finite exchange property implies the full exchange property, namely, modules whose special local summands are summands, and C4-modules for which the union of any chain of special summands is a summand. More precisely, we show that if either M is a module whose special local summands are summands or M is a C4-module such that the union of any chain of special summands of itself is a summand, then M has the finite exchange property if and only if it has the full exchange property if and only if it is a clean module. As immediate applications of the aforementioned results, we generalize and extend many of the fundamental results on the subject of exchange modules. [ABSTRACT FROM AUTHOR]

AbstractLet σ={σi|i∈I} be some partition of the set of all primes P, G a finite group and σ(G)={σi|σi∩π(G)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i and H contains exactly one Hall σi-subgroup of G for every σi∈σ(G). Let H be a complete Hall σ-set of G. A subgroup H of G is said to be H-permutable if HA = AH for every member A∈H; m- H-permutable if H=〈A,B〉 for some modular subgroup A and H-permutable subgroup B of G; weakly m- H-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H∩T≤S≤H for some m-H-permutable subgroup S of G. In this paper, we study the structure of finite groups G by assuming that some σ-primary subgroups are weakly m-H-permutable in G. Some recent results are generalized and unified. [ABSTRACT FROM AUTHOR]

AbstractLeibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

AbstractLet (C,E,s) be an extriangulated category with a proper class ξ of E-triangles. In this paper, we study Gorenstein derived functors for extriangulated categories. More precisely, we first introduce the notion of the proper ξ‐Gprojective resolution for an object in C and define the functors ξxtGP(ξ) and ξxtGI(ξ). Under some assumptions, we give some equivalent characterizations for ξ‐Gprojective and ξ‐Ginjective dimensions of objects in C by vanishing of such functors. As an application, our main results generalize Ren and Liu’s work. Moreover, our proof is not far from the usual module categories or triangulated categories. [ABSTRACT FROM AUTHOR]

AbstractIn a previous work, (dual)-m -Rickart lattices were studied. Now, in this paper, we introduce m -endoregular lattices as those lattices L such that m is a regular monoid, where m is a submonoid with zero of Endlin(L) . We show that these lattices can be characterized in terms of m -Rickart and dual-m -Rickart lattices. Also, we compare these new lattices with those lattices in which every compact element is a complement. We characterize the m -endoregular lattices such that every idempotent in m is central in m and we show that for these lattices the complements are a sublattice which is a Boolean algebra. We introduce two new concepts, m -K -extending and m -T -lifting lattices. For these lattices, we show that the monoid m has a regular quotient monoid provided they satisfy m -C2 and m -D2 respectively. At the end, some applications to quotients categories of R-Mod are given. [ABSTRACT FROM AUTHOR]

AbstractLet m, n be integers with m>n . Denote Mm,n by the set of all m×n matrices over the field F of characteristic zero. Let I be an n×m matrix with (i,i) -position 1 for any 1≤i≤n , and 0 in other position. Define a bracket [A,B]=AIB−BIA , where A,B∈Mm,n . Then Mm,n with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by gl(m×n,F) . In this paper, all derivations and biderivations of gl(m×n,F) are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on gl(m×n,F) are given. [ABSTRACT FROM AUTHOR]

AbstractWe define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G dividing the number of irreducible characters of G. In this paper, we classify odd-order nonabelian groups G with anz(G)≤2. [ABSTRACT FROM AUTHOR]

AbstractIt is known that the degree of irrationality irr(X) of a smooth surface X of degree d≥5 in P3 is at least d – 2. In this paper, we study a Galois rational map f:X−−→P2 of degree d – 2, and we show that the Galois group of f is a cyclic group, and f is defined by the family of lines intersecting two fixed skew lines. In addition, we give a necessary and sufficient condition that X has skew lines which give a Galois rational map of degree d – 2 by using automorphisms of X. [ABSTRACT FROM AUTHOR]