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This paper gives an explicit formula for the Hilbert series of algebras of Veronese type. [ABSTRACT FROM AUTHOR]

All groups considered in this paper are finite soluble groups. Let t p ( G ) = max H??G { j|p j ||| H G : H |} and t ( G ) = max p ?p( G ) t p ( G ). In this paper, we study the influence of t ( G ) on the structure of a finite soluble group G . In particular, the bounds of the nilpotent length (derived length, p -length) of a soluble group G with t ( G ) = 2 are found. [ABSTRACT FROM AUTHOR]

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, the structure of locally finite groups of infinite rank whose subgroups of infinite rank are permutable in a subgroup of finite index is investigated. [ABSTRACT FROM AUTHOR]

This paper is part of a larger programme investigating the B -injectors in arbitrary groups, more precisely, investigating in which groups the B -injectors are conjugate. To answer this question it is critical to have information on the intersection of the B -injectors with the components of the group. Here we are dealing with components having a factor group isomorphic to G 2 ( q ), 3 D 4 ( q ), S z ( q ), or R ( q ). As it is usually quite easy to determine the intersections of B -injectors with components corresponding to Chevally groups of large rank, the groups above among others turn out to be critical in answering the question whether B -injectors are conjugate. [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]

Neumann characterized the groups in which every subgroup has finitely many conjugates only as central-by-finite groups. If [Xfr] is a class of groups, a group G is said to have [Xfr]-conjugate classes of subgroups if G/CoreG(NG(H)) ∈ [Xfr] for every subgroup H of G. In this paper, we generalize Neumann's result by showing that a group has polycyclic-by-finite classes of conjugate subgroup if and only if it is central-by-(polycyclic-by-finite). [ABSTRACT FROM AUTHOR]

The groups having exactly one normalizer are well-known. They are the Dedekind groups. All finite groups having exactly two normalizers were classified by M. D. Pérez-Ramos and, in a recent paper, S. Camp-Mora generalized that result to locally finite groups. In this paper, we will characterize arbitrary groups with a finite number of normalizers and we will investigate the properties of arbitrary groups with two, three and four normalizers. [ABSTRACT FROM AUTHOR]

In this paper, we will determine the existence of blocks with sub-maximal subgroups of Sylow p-subgroups of a finite group G as defect groups. [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]

The main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G 2 ( q ), where q is a power of a prime p > 3. [ABSTRACT FROM AUTHOR]

A subgroup K of a finite group G is called an ℋ-subgroup of G if the following condition is satisfied: The set of all ℋ-subgroups of a finite group G will be denoted by ℋ( G ). In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of prime power orders belong to ℋ( G ). Communicated by M. Dixon [ABSTRACT FROM AUTHOR]

Let G be a group and let N be a normal subgroup of G. We set cd( G | N ) to be the degrees of the irreducible characters of G whose kernels do not contain N. We associate a graph with this set. The vertices of this graph are the primes dividing degrees in cd( G | N ), and there is an edge between p and q if pq divides some degree in cd( G | N ). In this paper, we study this graph when it is disconnected, and we study its diameter when it is connected. [ABSTRACT FROM AUTHOR]

The test rank tr( G ) of G is the minimum cardinality of a test set. In this paper we prove: I. Let N = N rc be a free nilpotent group of rank r = 2 and class c = 2. Then (i) tr( N ) = 2 for r odd and c = 2; (ii) tr( N ) = 1 in all other cases; (iii) an element g ? N 2 q , 2 is a test element if and only if it can be written as g = · s , where t 1 ,..., t q are nonzero integers. II. Let F r be a free group of rank r = 2 and A n be a free abelian group of rank n = 1. Then the group G = G rn = F r × A n has the test rank n . [ABSTRACT FROM AUTHOR]

In this paper we prove that a finite group G is isomorphic to the finite simple group L n ( q ) with n = 3 if and only if they have the same set of order of solvable subgroups. [ABSTRACT FROM AUTHOR]

In the paper, a Frattini-like subgroup associated with a class ?? of simple groups is introduced and analyzed. The corresponding ??-saturated formations are exactly the ??-local ones introduced by Förster. Our techniques are also useful to highlight the properties and behavior of ?-local formations. In fact, extensions and improvements of several results of Shemetkov are natural consequences of our study. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K 2 E is not generated by the Steinberg symbols { a , b } with a ? F *, b , ? E *. If however, F is a number field and D is a finite-dimensional central division F -algebra with square free index, then K 2 D is always generated by the Steinberg symbols { a , b } with a ? F *, b ? D *. Finally, the tame kernels of central division algebras over F are expressed explicitly. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study Lie algebras L such that if a subalgebra U of L has a maximal subalgebra of dimension one then every maximal subalgebra of U has dimension one. Such an L is called lm(0)-algebra. This class of Lie algebras emerges when it is imposed on the lattice of subalgebras of a Lie algebra the condition that every atom is lower modular. We see that the effect of that condition is highly sensitive to the ground field F. If F is algebraically closed, then every Lie algebra is lm(0). By contrast, for every algebraically non-closed field there exist simple Lie algebras which are not lm(0). For the real field, the semisimple lm(0)-algebras are just the Lie algebras whose Killing form is negative-definite. Also, we study when the simple Lie algebras having a maximal subalgebra of codimension one are lm(0), provided that char(F) ≠ 2. Moreover, lm(0)-algebras lead us to consider certain other classes of Lie algebras and the largest ideal of an arbitrary Lie algebra L on which the action of every element of L is split, which might have some interest by themselves. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the structure of finite groups G admitting a normal subgroup N such that N contains almost all conjugacy classes of G. [ABSTRACT FROM AUTHOR]

In this paper, we show that σ-reducibility is preserved under joins with K. where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with a the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that σ- reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product. we prove that SI K is ,κ-tame, where SI stands for the pseudovariety of semilattiees. [ABSTRACT FROM AUTHOR]

Given a numerical semigroup S. let M(S) = S\{0} and (IM(S) - IM(S)) - {x ϵ No : x · IM(S) ⊆ IM(S)}. Define associated numerical semigroups B(S) := (M(S) - M(S)) and L(S) := Ut=i∞ (IM(S) - IM(S)). Set B0(S) - S, and for i ≥ 1, define Bi(S) :- B(Bi-1 (S). Similarly, set L0(S) - S, and for i ≤ 1, define Li(S) :- L(Li-1(S)). These constructions define two finite ascending chains of numerical semigroups S = B0(S) ⊆ B1(S) ⊆ … ⊆ BβS(S) - N0 and S - L0(S) ⊆ L1(S) ⊆ … ⊆ Lλ(S) = N0. It has been shown that not all numerical semigroups S have the property that Bi(S) ⊆ Li(S) for all i ≥ 0. In this paper, we prove that if S is a numerical semigroup with a set of generators that from a generalized arithmetic sequence, then Bi(S) ⊆ Li(S) for all i ≥ 0. Moreover, we see that this containment is not necessarily satisfied if a set of generators of S form an almost arithmetic sequence. In addition, we characterize numerical semigroups generated by generalized arithmetic sequences that satisfy other semigroup properties, such as symmetric, pseudo-symmetric, and Arf. [ABSTRACT FROM AUTHOR]

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184-193]. It was proved that PSL(3,q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3,q) where q is an odd prime power. J. Pure Appl. Algebra 170(2-3): 243-254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3,q) where q = 2n. Acta Math. Sinica, English Ser. 18(3):463-472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5,q). Bull. Austral. Math. Soc. 65:211-222] it was proved that PSL(3,q) for q = 2n and PSL(5,q) are uniquely determined by their order components. In this paper we prove that PSL(p,q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration. [ABSTRACT FROM AUTHOR]

A subgroup of a group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be S-quasinormally embedded in G if every Sylow subgroup of H is a Sylow subgroup of some S-quasinormal subgroup of G. In this paper we examine the structure of a finite group G under the assumption that certain abelian subgroups of prime power order are S-quasinormally embedded in G. Our results improve and extend recent results of Ramadan [Ramadan, M. (2001). The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups. Arch. Math. 77:143-148]. [ABSTRACT FROM AUTHOR]

In this paper we are interested in the relationship between the Wielandt series and strong Wielandt series of a finite soluble group. Some exact bounds on strong Wielandt length in terms of Wielandt length are obtained for groups with p-length 1 for all primes p. We also give two new results relating the Wielandt subgroup and strong Wielandt subgroup of an arbitrary finite soluble group. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to construct an "algebraic" representation of a group automorphisms Aut F for any elementary solvable group Γ. "Algebraic" means that the image of a semisimple (unipotent) automorphism, in the sense of Wang, will be a semisimple (unipotent) matrix Theorem 1. This gives an answer on the question asking by Wang. As a corollary of this theorem, we show that for any group of automorphisms Aut Γ of a lattice F in a solvable connected group Lie G there exist a representation ρ: Aut Γ → GLn(Z), such that ρ(Aut Γ) is an arithmetic subgroup in the Zariski closure ρ(Aut Γ). [ABSTRACT FROM AUTHOR]

We give a complete description of primitive linear representations of a p-group over a field F of characteristic ≠2, p, including whether they are of orthogonal or symplectic type. For p odd this is easy and essentially well-known. For p=2, although the proofs offer no special difficulty the discussion is quite involved, depending on the cyclotomy of F. When F is non-exceptional in the sense of Harris and Segal, primitive representations can be obtained from "generic" ones (Theorems 9.8 and 10.4): this may be considered as the main result of the paper. In passing, we improve an old result of O. Schilling on the division ring of endomorphisms of an irreducible representation of a p-group. [ABSTRACT FROM AUTHOR]

In the present paper we consider a correspondence weaker than Galois connection and prove that this produces Kurosh-Amitsur radicals in a very general setting including all universal classes of O-groups. As a framework we introduce a simple combinatorial structure which uses mappings between complete lattices. [ABSTRACT FROM AUTHOR]

This work was intended as an attempt to continue the study of the class &scriptB; of generalised nilpotent groups started in a previous paper. We present some results concerning the Fitting subgroup and the &scriptB;-injectors of a radical locally finite group satisfying min-p for all p. [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]

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 identify Čech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L ∞-algebra. Applications to deformation theory are described. [ABSTRACT FROM PUBLISHER]

Using a unified method, we determine the structure of automorphisms and representations of arbitrary polyadic groups. More precisely, for a polyadic group (G, f) = der θ, b (G, ·), we obtain a complete description of automorphisms and representations of (G, f) in terms of automorphisms and representations of the binary group (G, ·). [ABSTRACT FROM PUBLISHER]

We study the restriction of the Weil representations of symplectic and unitary groups to subgroups that are the centralizers of certain elements, and show that these are multiplicity free. This work is along the line of the Howe philosophy for so-called dual pairs in the symplectic and unitary groups, but our dual pairs are not reductive. We also obtain an explicit formula for the character of the restriction to the centralizer of a regular unipotent element. [ABSTRACT FROM AUTHOR]

The goal of this article is to study finite groups admitting a pseudocomplemented subgroup lattice (PK-groups) or a pseudocomplemented normal subgroup lattice (PKN-groups). In particular, we obtain a complete classification of finite PK-groups and of finite nilpotent PKN-groups. We also study groups with a Stone normal subgroup lattice, and we classify finite groups for which every subgroup has a Stone normal subgroup lattice. Finally, we obtain a complete classification of finite groups for which every subgroup is monolithic. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called an NR - subgroup (Normal Restriction) if whenever K ⊴ H, then KG ∩ H = K, where KG is the normal closure of K in G. In this article, we will prove some sufficient conditions for the solvability of finite groups which possess many NR-subgroups. We also prove a criterion for the existence of a normal p-complement in finite groups. [ABSTRACT FROM AUTHOR]

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. We obtain two new lower bounds for ν(G) when G is a non-abelian finite p-group and p is odd. More precisely, if |G| =pn, exp Z(G) = pe, and exp G/G' =pf, let us define λ(G) = n - e and κ(G) = n - f. Then we prove that ν(G) ≥ p(λ(G) -3) +2 and ν(G) ≥ p(κ(G) -3) +2. The first bound improves the bound ν(G) ≥ λ(G) -1 given by [10], and almost in every case, the second one improves the bound ν(G) ≥ p(k - 1) +1 obtained by [6], where k is defined by the condition that |G'| =pk. [ABSTRACT FROM AUTHOR]

Given a ring R and a class C of R-modules, one can ask whether or not every element of C decomposes uniquely as a direct sum of indecomposable elements of C. If not, one can further ask if it is possible for an element of C to decompose both as the direct sum of s indecomposable elements of C and as the direct sum of t indecomposable elements of C where s ≠ t. In this article, we investigate these questions when R is a two-dimensional analytically normal domain and C is the class of finitely generated torsion-free R-modules. [ABSTRACT FROM AUTHOR]

For any locally inverse semigroup S, there exists a maximal dense ideal extension of S within the class LI of all locally inverse semigroups (Pastijn and Oliveira, 2006, Preprint). Here we realize this maximal dense ideal extension in terms of a canonically constructed quotient of a regular Rees matrix semigroup over an inverse semigroup. [ABSTRACT FROM AUTHOR]

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ⊗ E(A)A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in G, the class of self-small mixed groups G with G/tG≅ n for some n < ω. We obtain a new characterization of the elements of G, and demonstrate that G differs in various ways from the class TF of torsion-free abelian groups of finite rank despite the fact that the quasi-category G is dual to a full subcategory of TF. [ABSTRACT FROM AUTHOR]

We prove that, if H is a finite-dimensional semisimple Hopf algebra, and A is an FCR H-module algebra over an algebraically closed field, then A is a PI-algebra, provided the subalgebra of invariants is a PI-algebra. We also show that if A is an affine algebra with an action of a finite group G by automorphisms, the subalgebra of the fixed points AG is in the center of A, and the characteristic of the ground field is either zero or relatively prime to the order of G, then AG is affine. Analogous results are proved for graded algebras and H-module algebras over a semisimple triangular Hopf algebra over a field of characteristic zero. We prove also that, if A is an H-module algebra with an identity element, and H is either a semisimple group algebra or its dual, then, if A is semiprimitive (semiprime), then so is AH. [ABSTRACT FROM AUTHOR]

The skeleton of the lattice of all structurally trivial semigroup varieties is known to be isomorphic to an infinitely ascending inverted pyramid (Kopamu, 2003). We digitize the skeleton by representing each variety forming the skeleton as an ordered triple of non-negative integers. This digitization of the lattice, under the pointwise ordering of non-negative integers, provides useful algorithms which could easily be programmed into a computer, and then used to compute varietal joins and meets, or even to draw skeleton lattice diagrams. An application to a certain larger subvariety lattice is also given as an example. [ABSTRACT FROM AUTHOR]

Let G be an Abelian group whose p-component Gp is summable and let F be a field of characteristic p. Let S(FG) be the normalized unit p-group of the group algebra FG. The main results of the present article are that Gp is a direct factor of S(FG) with totally projective complement, provided Gp is of countable length and F is perfect; and, in particular, under these circumstances S(FG) is summable if and only if Gp is summable. Moreover, if FG≅ FH as F-algebras for some group H and if Gp is summable, then Hp is summablale. These achievements improve results due to Hill and Ullery (1997) and also their generalizations given by Danchev (2000a, b, 2001, 2004). [ABSTRACT FROM AUTHOR]

In this article we investigate the Bol loops and connected with them groups. We prove an analog of the Doro's theorem for Moufang loops and find a criterion for simplicity of Bol loops. One of the main results obtained is the following: If the right multiplication group of a connected finite Bol loop S is a simple group, then S is a Moufang loop. [ABSTRACT FROM AUTHOR]

Given an infinite-dimensional vector space V , we consider the semigroup KN ( p , q ) consisting of all injective linear transformations α : V → V , for which the codimension of the range of α is at least q , where dim V = p ≥ q ≥ ℵ 0 . Kemprasit and Namnak (2001) considered the semigroup KN ( p , ℵ 0 ) while deciding when certain subsemigroups of T ( V )—the semigroup under composition of all linear transformations from V to V —belong to BQ—the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. In this article, we determine when KN ( p , q ) belongs to BQ in terms of the dimension of V . Next we characterize Green's relations on KN ( p , q ), and determine its one and two sided ideals; and we use this information to show that KN ( p , q ) is a model for certain types of algebraic semigroups. Then we describe all quasi-ideals and bi-ideals of KN ( p , q ). We also determine its maximal right simple subsemigroups. Communicated by D. Easdown. [ABSTRACT FROM AUTHOR]

We use the symmetric product of the projective line to describe the resultant scheme R n , m in as a quotient X / G where and G ⊆ Aut k ( X ) is a finite subgroup. As a special case we give a description of the discriminant scheme in terms of the symmetric product. [ABSTRACT FROM AUTHOR]

Let F m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m , with m ≥ 2. We show that F m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F (N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F (N). We prove that the fixed point subalgebra F (N) G is not finitely generated. [ABSTRACT FROM AUTHOR]

We prove that the Baer-Specker group Π = Zχ0 contains a pure subgroup isomorphic to the direct sum of 2χ0 copies of itself. We produce 22χ0 nonisomorphic subgroups of n, each isomorphic to its dual group. Finally, we show that the isomorphism type of a generalized product of Z's, the set of functions I ↵ Z with support of size at most α, uniquely determines both the cardinality of I and a (as long as there are no measurable cardinals ≤α). Alt three of these results are obtained using set-theoretic existence theorems, namely, the existence of large independent families, large almost disjoint families, and Δ-systems. [ABSTRACT FROM AUTHOR]

A ring R is called an n-clean (resp. S-clean) ring if every element in R is n-clean (resp. S-clean). Clean rings are 1-clean and hence are S-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that S-clean rings are a proper generalization of clean rings. The group ring Z ( p ) G with G a cyclic group of order 3 is proved to be S-clean. The m× m matrix ring M m ( R ) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M m (R) over any ring is S-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian p-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a 1 R + ??? a m R = dR, with m = 2, a 1 ,..., a m , d ? R, there exist u 1 ? U(R) and u 2 ,..., u m ? W(R) such that a 1 u 1 + c a m u m = Rd. [ABSTRACT FROM AUTHOR]

The aim of this work is to describe the irreducible components of the nilpotent complex Leibniz algebras varieties of dimension less than 5. We construct degenerations between one-parametric families of nilpotent Leibniz algebras and study the rigidity of these families. [ABSTRACT FROM AUTHOR]

If G is a nontrivial finite group with the property that ?(1) 2 divides bi | G | for all irreducible (complex) characters ?, then G has a nontrivial normal abelian subgroup. [ABSTRACT FROM AUTHOR]