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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]

Letbe a connected reductive algebraic group defined over an algebraically closed field. Letbe a maximal torus ofand letbe a Borel subgroup ofcontaining. LetWdenote the Weyl group ofand letẇdenote an element in the preimage of the natural map fromtoW. We investigate what conditions are necessary and sufficient for the double cosets(called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras. [ABSTRACT FROM PUBLISHER]

We show that if |Aut(G)| is odd, then without loss of generality we may assume the only primes dividing |G| are those dividing |Aut(G)|. We also classify finite abelian groups A such that |Aut(A)| = |A| and finite abelian groups A such that |Aut(A)| p = |A| p , where p is a prime dividing |A|. [ABSTRACT FROM PUBLISHER]

Let G be a finite group and ℱ a formation. A subgroup H is called ℱ-quasinormal in G if there exists a quasinormal subgroup T of G such that HT is quasinormal in G and (H ∩ T)H G /H G is contained in the ℱ-hypercenter of G/H G . In this article, we study the structure of finite groups by using ℱ-quasinormal subgroups and prove that: Let ℱ be a saturated formation containing 𝒰 and G be a group with a normal subgroup H such that G/H ∈ ℱ. If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is 𝒰-quasinormal in G, then G ∈ ℱ. [ABSTRACT FROM PUBLISHER]

The aim of this work is to provide an algebraic-geometric method to construct generalized local symbols on curves as morphisms of group schemes. From a closed point of a complete, irreducible, and nonsingular curve C over a perfect field k as the only data, using theta groups over Picard schemes of curves, we offer a geometric construction that allows us to define generalizations of the tame symbol and the Hilbert norm residue symbol. [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]

We classify the irreducible complex characters of the symplectic groups Sp2n(q) and the orthogonal groups [image omitted], Spin2n+1(q) of degrees up to the bound D, where D = (qn - 1)q4n-10/2 for symplectic groups, D = q4n-8 for orthogonal groups in odd dimension, and D = q4n-10 for orthogonal groups in even dimension. [ABSTRACT FROM AUTHOR]

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves. [ABSTRACT FROM AUTHOR]

Given two affine algebraic varieties X and Y such that X ×  ≅ Y × , it may happen that X and Y are not isomorphic. This is shown, for example, by so-called “Danielewski surfaces.” In this note, we give a simple construction of such varieties X, Y of an arbitrary dimension n > 1. [ABSTRACT FROM AUTHOR]

For any semigroup S a regular semigroup C(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schutzenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S. [ABSTRACT FROM AUTHOR]

We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect, and quasi-adequate transversal and provide a number of interesting connections between these. [ABSTRACT FROM AUTHOR]

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F)/G. In this article, we classify the surfaces of general type with pg = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out that they belong to four families, that we call surfaces of type I, II, III, IV. The moduli spaces I, II, IV are irreducible, whereas III is the disjoint union of two irreducible components. In the last section we start the analysis of the case where G is not abelian, by constructing several examples. [ABSTRACT FROM AUTHOR]

We show that there exists an essentially finitely indecomposable abelian p-group which is not a thick group, thus resolving once again in the negative an Irwin's conjecture which was already settled by Dugas and Irwin (1992) and Danchev (2008). Some other related questions are studied as well. [ABSTRACT FROM AUTHOR]

We introduce and study the Picard group of a coring. We give an exact sequence relating the Picard group of a coring and its automorphisms generalizing the known exact sequences associated to an algebra and a coalgebra over a field. We also extend to corings the Aut-Pic property and we give some new corings satisfying this property. [ABSTRACT FROM AUTHOR]

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established. [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]

This article provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive the description of the natural partial order of unipotent classes in the general linear group in terms of their Jordan blocks. This result was originally due to Gerstenhaber. Finally, we obtain a proof of the Bala-Carter Theorem, which holds even in certain bad characteristics. This proof requires the prior knowledge of the number of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening. [ABSTRACT FROM AUTHOR]

L-functors (Rump, 2001) provide a new tool for the study of Auslander--Reiten quivers associated with an isolated singularity in the sense of M. Auslander. We show that L-functors L, L- : M → M admit an intrinsic definition for an arbitrary additive category M. When they exist, they endow M with a structure closely related to that of a triangulated category. If M is the homotopy category M(&scriptA; ) of two-termed complexes over an additive category &scriptA;, we establish a one-to-one correspondence between L-functors on M(&scriptA;) and classes of short exact sequences in &scriptA; which make &scriptA; into an exact category with almost split sequences. This applies, in particular, to categories, &scriptA; = Λ-CM of Cohen--Macaulay modules over a Cohen--Macaulay R-order Λ for arbitrary dimension of R. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ϵ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type. [ABSTRACT FROM AUTHOR]

We study the automorphism and collineation groups of plane curves, i.e., in IP², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational. [ABSTRACT FROM AUTHOR]

In this article we prove Broué's conjecture on isotypies for the case of principal blocs of Sp(4, Fq)〈σ〉 : a split cyclic extension of the symplectic group of dimension 4 over a finite field Fq for q odd. The core of the proof is to lift to the group Sp(4, Fq)〈σ〉 an isotypy of the group Sp(4, Fq). For this purpose, we build a convenient version of a theorem of lifting isotypies. [ABSTRACT FROM AUTHOR]