Algebra Colloquium

The Orlik–Solomon algebra is a graded algebra defined by the partially ordered set of subspace intersections of the hyperplanes in an arrangement. Define the cohomology of an Orlik–Solomon algebra as that of the complex formed by its homogeneous components with the differential defined via multiplication by an element of degree one. The dimension of the cohomology of the Orlik–Solomon algebra in dimension one has been determined by Libgöber and Yuzvinsky. Using similar techniques, we study the dimension of the cohomology groups of the Orlik–Solomon algebra in higher dimensions under the special case where the element of degree one which defines the multiplication is concentrated under an element of the intersection lattice of codimension two. We provide computational methods for the dimension of the second cohomology group.

In this paper, we classify metacyclic p-groups of nilpotency class at least three up to isomorphism by giving a canonical presentation for each isomorphism class.

We prove that if F is a finite group, and G and H are finitely generated groups with finite commutator subgroups for which F × G ≃ F × H, then G ≃ H.

If G is a finite group and is a group of automorphisms of G, then it is known that the matrix near-ring is a subnear-ring of the centralizer near-ring for every m ≥ 2. Conditions are known under which is a proper subnear-ring of , and if and G are abelian, then conditions are known which imply the equality . In this paper, we characterize the groups of automorphisms of a cyclic p-group G for which this equality holds. We also show that for every group of automorphisms of a cyclic p-group G, either all the nonzero orbits of G are of unique type or none of the orbits of G is of unique type if p is odd, and there is a third possibility if p=2, namely precisely one of the orbits of G is of unique type.

In this paper, we show that the coset diagram for the action of a linear-fractional group generated by and on the rational projective line is transitive and the coset diagram for the action is connected. Using the coset diagram, we show that are defining relations for the group.

Let be a semilattice Ω-graded ring and P a radical. Conditions are found under which if and only if for all α, β ∈ Ω. Examples are given to show that some conditions cannot be removed.

Let F be a free group and R be a subgroup of F. It is proved that

Let v be a henselian valuation of a field K, and be the (unique) extension of v to a fixed algebraic closure of K. For an element , a chain of elements of such that θ_i is of minimal degree over K with the property that and θ_m ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ.

Let K be a complete field with respect to a real non-trivial valuation v, and be the extension of v to an algebraic closure of K. A well-known result of Ostrowski asserts that the limit of a Cauchy sequence of elements of does not always belong to unless is a finite extension of K. In this paper, it is shown that when a Cauchy sequence { b_n } of elements of is such that the sequence { [K(b_n) : K] } of degrees of the extensions K(b_n) / K does not tend to infinity as n approaches infinity, then { b_n } has a limit in . We also give a characterization of those Cauchy sequences { b_n } of elements of whose limit is not in , which generalizes a result of Alexandru, Popescu and Zaharescu.

We extend the five-term exact sequence of homology with trivial coefficients of Leibniz n-algebras associated to a central extension of Leibniz n-algebras by means of a sixth term which is a generalization of the Ganea term for homology of Leibniz algebras. We use this sequence in order to analyze several questions related with the centre and central extensions of a Leibniz n-algebra.

In this paper, we prove that a binary sequence is perfect (resp., quasi-perfect) if and only if its support set for any finite group (not necessarily cyclic) is a Hadamard difference set of type I (resp., type II); and we prove that the kernel of any nonzero linear functional (or the image of any linear transformation A with dim (KerA) = 1) on the linear space GF(2^m) over the field GF(2) (excluding 0) is a cyclic Hadamard difference set of type II using Gaussian sums; and we prove that the multiplier group of the above difference set is equal to the Galois group Gal (GF(2^m) / GF(2)); and we mention the relationship between the Hadamard transform and the irreducible complex characters.

A natural development of the recent work about semistar operations leads to investigate the concept of “semistar invertibility”. Here, we show the existence of a “theoretical obstruction” for extending many results, proved for star-invertibility, to the semistar case. For this reason, we introduce two distinct notions of invertibility in the semistar setting (called ⋆-invertibility and quasi-⋆-invertibility), we discuss the motivations of these “two levels” of invertibility and we extend, accordingly, many classical results proved for the d-, v-, t- and w-invertibility.

In the context of G-algebras, we prove that Green correspondent points satisfy some important properties that are suggested by the classical finite group Green correspondence.

Let G be a finite group. A subgroup H of G is called conjugate-permutable in G if HH^g = H^gH for any g ∈ G. A group G is called an ECP-group if every subgroup of G is conjugate-permutable in G. In this paper, we study the influence of conjugate-permutable subgroups on the structure of a finite group, especially on the nilpotency or supersolvability of the group, and give some sufficient or necessary conditions for a finite group to be an ECP-group.

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

A cover for a finite group is a set of proper subgroups whose union is the whole group. For a finite group G, we denote by σ (G) the least positive integer n such that G has a cover of size n. This paper deals with the covers of a finite soluble group G having σ (G) elements of maximal subgroups.

Let G be a finite solvable group. We prove that any prime dividing any irreducible π-partial character degree of G divides the size of some conjugacy class of π-elements of G. Under certain hypothesis, we show that if two distinct primes r and s both divide some irreducible π-partial character degree, then there exists a conjugacy class of π-elements whose size is divisible by rs.

If K is a set of automorphisms of a group G, an endomorphism θ : G → G is said to be K-pointwise if for each element t ∈ G, there exists an element φ ∈ K such that θ (t) = φ (t). This generalizes the notion of pointwise inner automorphism. We show that in some special cases, a K-pointwise endomorphism is necessarily an automorphism (it is not true in general).

In this paper, some new modules over the rank two quantized Weyl algebra are presented. The isomorphism classes among these modules are also determined.