A note on fuzzy HV-submodules

The purpose of this paper is to present certain results arising from product between fuzzyH _v -submodules. In particular, we consider the fundamental relatione ^* defined on anH _v -module and give a property of the fundamental relations and fundamental modules with respect to the fuzzy product ofH _v -modules.     

Some properties ofTL-groups

We introduce the notion ofTL-p-subgroups that is an extension of the notion of fuzzyp-subgroups and show that a torsionTL-subgroup of an Abelian group withT = Λ can be written as the intersection of its minimalTL-p-subgroups.     

Construction of Ternary H v -groups and Ternary P-hyperoperations

In this paper, we study the concept of ternary H _v -groups and some their properties. We give some examples of ternary H _v -groups. Also, we consider the fundamental relation β* on a ternary H _v -group and prove that β* is a compatible relation on a ternary H _v -group. In addition, we define the P-hyperoperation, and then, we construct a new ternary H _v -group.     

Product of fuzzyH υ-ideals inH υ-rings

In this paper we define product between fuzzyH _υ-ideals of givenH _υ-rings. We consider the fundamental relation γ* defined on anH _υ-ring and give some properties of the fundamental relations and fundamental rings with respect to the product of fuzzyH _υ-ideals.     

Fundamental relation and automorphism group of very thin Hv-groups

The purpose of this paper is computing the fundamental relations and automorphism groups of very thin H_v-groups. In this regards, we first investigate some basic properties of H_v-groups and then we show that any given group is isomorphic to the fundamental group of a nontrivial H_v-group. The main properties of very thin H_v-groups are also investigated and it is proved that every finite very thin H_v-group is proper.     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.     

On the strongly closed subgroups or H-subgroups of finite groups

Let G be a finite group. Goldschmidt, Flores, and Foote investigated the concept: Let K ≤ G. A subgroup H of K is called strongly closed in K with respect to G if H ^g ∩ K ≤ H for all g ∈ G. In particular, when H is a subgroup of prime-power order and K is a Sylow subgroup containing it, H is simply said to be a strongly closed subgroup. Bianchi and the others called a subgroup H of G an H-subgroup if N _G (H) ∩ H ^g ≤ H for all g ∈ G. In fact, an H-subgroup of prime power order is the same as a strongly closed subgroup. We give the characterizations of finite non-T-groups whose maximal subgroups of even order are solvable T-groups by H-subgroups or strongly closed subgroups. Moreover, the structure of finite non-T-groups whose maximal subgroups of even order are solvable T-groups may be difficult to give if we do not use normality.     

A Generalization of Cover-Avoiding Properties in Finite Groups

A subgroup H of a group G is called a CAP*-subgroup of G, if H either covers or avoids every non-Frattini chief factor of G. In this article, we give some interesting properties of CAP*-subgroups. Furthermore, we determine the structure of finite groups based on the assumption that some subgroups are CAP*-subgroups and generalize some known results.     

Finite groups with some NR-subgroups or {\mathcal{H}}-subgroups

Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if whenever ${K \trianglelefteq H}$ , then ${K^G \cap H = K}$ , where K ^G is the normal closure of K in G. Bianchi, Gillio Berta Mauri, Herzog and Verardi proved a characterization of soluble T-groups by means of ${\mathcal{H}}$ -subgroups: a subgroup H of G is said to be an ${\mathcal{H}}$ -subgroup of G if ${H^g \cap N_G(H) \leq H}$ for all ${g \in G}$ . In this article we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or ${\mathcal{H}}$ -subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.     

Deductive systems of a cone algebra — II: Isomorphism theorem

We prove that there is an isomorphism φ of the lattice of deductive systems of a cone algebra onto the lattice of convex ℓ-subgroups of a lattice ordered group (determined by the cone algebra) such that for any deductive system A of the cone algebra, A is respectively a prime, normal or polar if and only if φ(A) is a prime convex ℓ-subgroup, ℓ-ideal or polar subgroup of the ℓ-group, thus generalizing and extending the result of Rachůnek that the lattice of ideals of a pseudo MV-algebra is isomorphic to the lattice of convex ℓ-subgroups of a unital lattice ordered group.      

On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if N _G (H) ∩ H ^x  ≤ H for all x ∈ G. A subgroup H of G is called weakly ?-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ?-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ?-subgroups in G. Some recent results are extended and generalized.     

On <Emphasis Type="Italic">π</Emphasis>-S-permutable subgroups of finite groups

Let π be a set of primes. A subgroup H of a finite group G is said to be π-S-permutable in G if H permutes with every Sylow q-subgroup of G for all primes q ∈ π. The main aim of this paper is to establish structural results about the normal closure of π-S-permutable subgroups and p-subgroups permuting with all p′-subgroups for a single prime p. Our results stem from a recent article by Isaacs [5] and subsequent discussions with the authors about it.     

On the Pronormality of Subgroups of Odd Index in Some Extensions of Finite Groups

We study finite groups with the following property (*): All subgroups of odd index are pronormal. Suppose that G has a normal subgroup A with property (*), and the Sylow 2-subgroups of G/A are self-normalizing. We prove that G has property (*) if and only if so does N_G(T)/T, where T is a Sylow 2-subgroup of A. This leads to a few results that can be used for the classification of finite simple groups with property (*).     

Isotype knice subgroups of global Warfield groups

If H is an isotype knice subgroup of a global Warfield group G, we introduce the notion of a k-subgroup to obtain various necessary and sufficient conditions on the quotient group G/H in order for H itself to be a global Warfield group. Our main theorem is that H is a global Warfield group if and only if G/H possesses an H(ℵ₀)-family of almost strongly separable k-subgroups. By an H(ℵ₀)-family we mean an Axiom 3 family in the strong sense of P. Hill. As a corollary to the main theorem, we are able to characterize those global k-groups of sequentially pure projective dimension ⩽ 1.     

Sheaves of Abelian l-groups

Keimel was first to associate a spectrum with an abelian l-group G. An alternative construction of a spectrum is proposed here. Its points are the prime l-ideals of G, i.e., they are the points of Keimel’s spectrum plus the improper prime l-ideal G. The spectrum is equipped with the inverse topology, which is different from the one used by Keimel. The new spectrum is denoted by Spec_inv(G) and is called the inverse spectrum of G. There is a natural construction of a sheaf of l-groups on Spec_inv(G), which is called the structure sheaf of G on Spec_inv(G). The inverse spectrum and its structure sheaf first appeared in Rump and Yang (Bull Lond Math Soc 40:263–273, 2008) via the Jaffard-Ohm Theorem. We present an elementary and direct construction. The relationship between an l-group H and an l-subgroup G is analyzed via the structure sheaves on the inverse spectra. The structure sheaves are used to give a sheaf-theoretic presentation of the valuation theory of a field. The group of Cartier divisors of an integral affine scheme is identified canonically with an l-group. Special attention is paid to Prüfer domains, where the ties between rings and l-groups are particularly close.     

On the regular Sylow p-subgroups of Chevalley groups over \mathbb{Z}_{p^m }

We prove that a Sylow p-subgroup of the general linear group of dimension n over the residue ring modulo p ^m is regular for n ² < p and powerful if and only if n = 2 and m = 1. We obtain similar results for the Sylow p-subgroups of normal types over the same ring.     

An L-Point Characterization of Normality and Normalizer of an L-Subgroup of an L-Group

In this paper, we study the notion of normal $L$-subgroup of an $L$-group and provide its characterization by an $L$-point. We also provide a construction of the normalizer of an $L$-subgroup of a given $L$-group by using $L$-points. Moreover, we also discuss the product, homomorphic images and homomorphic preimages of normalizers.     

Exklusive Graphen und Hamiltonche Graphenn-ten Grades II

The graphs considered are finite and undirected, loops do not occur. An induced subgraphI of a graphX is called animitation ofX, if

1. the degreesd _I(v)≡d _X(v) (mod 2) for allv∈V(I)
2. eachu∈V(X)?V(I) is connected with the setv(I) by an even number of edges. If the set of imitations ofX consists only ofX itself, thenX is anexclusive graph. AHamiltonian graph of degree n (abbr.:HG ⁿ) in the sense ofA. Kotzig is ann-regular graph (n>1) with a linear decomposition and with the property, that any two of the linear components together form a Hamiltonian circuit of the graph.

In the first chapter some theorems concerning exclusive graphs and Euler graphs are stated. Chapters 2 deals withHG ^n′ s and bipartite graphs. In chapters 3 a useful concept—theX-graph of anHG ⁿ—is defined; in this paper it is the conceptual connection between exclusive graphs andHG ^n′ s, since a graphG is anHG ⁿ, if all itsX-graphs are exlusive. Furthermore, some theorems onX-graphs are proved. Chapter 4 contains the quintessence of the paper: If we want to construct a newHG ⁿ F from anotherHG ⁿ G, we can consider certain properties of theX-graphs ofG to decide whetherF is also anHG ⁿ.     

List coloring of Cartesian products of graphs

A well-established generalization of graph coloring is the concept of list coloring. In this setting, each vertex v of a graph G is assigned a list L(v) of k colors and the goal is to find a proper coloring c of G with c(v)∈L(v). The smallest integer k for which such a coloring c exists for every choice of lists is called the list chromatic number of G and denoted by χ_l(G).We study list colorings of Cartesian products of graphs. We show that unlike in the case of ordinary colorings, the list chromatic number of the product of two graphs G and H is not bounded by the maximum of χ_l(G) and χ_l(H). On the other hand, we prove that χ_l(G×H)?min{χ_l(G)+col(H),col(G)+χ_l(H)}-1 and construct examples of graphs G and H for which our bound is tight.