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.     

Commutative Rings Obtained from Hyperrings (H v -rings) with α∗-Relations

The main tools in the theory of hyperstructues are the fundamental relations. The fundamental relation on a hyperring was introduced by Vougiouklis at the fourth AHA congress. The fundamental relation on a hyperring (H _v -ring) is defined as the smallest equivalence relation so that the quotient would be the (fundamental) ring. Note that the commutativity with respect to both sum and product in the (fundamental) ring are not assumed. Now, in this article we would like the (fundamental) ring to be commutative with respect to both sum and product, that is, the fundamental ring should be an ordinary commutative ring. Therefore we introduce a new strongly regular equivalence relation on hyperrings (H _v -rings). If we consider this relation on a hperring (H _v -ring), then the set of quotients is a commutative ring. Some properties of such rings are investigated.     

Dieudonné Completion and PT-Groups

We consider the classes of PT-groups, strong PT-groups, completion friendly groups, and Moscow groups introduced by Arhangel’skii for the study of the Dieudonné completion of topological groups. We show that every subgroup H of a Lindel?f P-group is a PT-group, and that H is a strong PT-group iff it is \mathbb R{\mathbb R}-factorizable. Assuming CH, we prove that every ω-narrow P-group is a PT-group. Several results regarding products of PT-groups and \mathbb R{\mathbb R}-factorizable groups are established as well. We prove that the product of a Lindel?f group and an arbitrary subgroup of a Lindel?f Σ-group is completion friendly, and the same conclusion is valid for the product of an \mathbb R{\mathbb R}-factorizable P-group with an almost metrizable group.     

Finite groups with some pronormal subgroups

A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H ^x in 〈H, H ^x 〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups.     

Finite groups in which permutability is a transitive relation on their frattini factor groups

Let G be a finite group. A PT-group is a group G whose subnormal subgroups are all permutable in G. A PST-group is a group G whose subnormal subgroups are all S-permutable in G. We say that G is a PT_o-group (respectively, a PST_o-group) if its Frattini quotient group G/Φ(G) is a PT-group (respectively, a PST-group). In this paper, we determine the structure of minimal non-PT_o-groups and minimal non-PST_o-groups.      

On the cohomological characterization of real free pro-2-groups

The aim of this note is to give a cohomological characterization of the real free pro-2-groups. Thereal free pro-2-groups are the free pro-2-product of copies of ℤ/2ℤ with a free pro-2-group. They are characterized as the pro-2-groupsG for which there exists a character χ₀, whose kernel is a free pro-2-group, such that χ₀∪χ=χ∪χ, for every χ∈H ¹(G). We discuss the naturalness of these conditions and we state some relations between them and field arithmetic properties. Supported by a grant from CNPq-Brasil. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Determinazione deiT 2-gruppi finiti

Summary A T-group is a group in which normality is transitive, a T₁-group is a group which is not a T-group but all of whose proper subgroups are T-groups, and a T₂-group is a group which is not a T- group or a T₁-group but all of whose proper subgroups are T-groups or T₁-groups. In this note we determine all the finite T₂-groups, and we show that all T₂-groups which are either soluble or 2-groups are finite. We study also the groups in which every proper soluble subgroup is a T-group or a T₁-group.

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. (C.N.R.).     

Almost Butler groups

Generalizing the notion of the almost free group we introduce almost Butler groups. An almost B ₂-group G of singular cardinality is a B ₂-group. Since almost B ₂-groups have preseparative chains, the same result in regular cardinality holds under the additional hypothesis that G is a B ₁-group. Some other results characterizing B ₂-groups within the classes of almost B ₁-groups and almost B ₂-groups are obtained. A theorem of [BR] stating that a group G of weakly compact cardinality having a -filtration consisting of pure B ₂-subgroup is a B ₂-group appears as a corollary.     

On the Schur multipliers of finite p-groups of given coclass

In this paper we obtain bounds for the order and exponent of the Schur multiplier of a p-group of given coclass. These are further improved for p-groups of maximal class. In particular, we prove that if G is p-group of maximal class, then |H ₂(G, ℤ)| < |G| and expH ₂(G, ℤ) ≤ expG. The bound for the order can be improved asymptotically.     

Groups of Automorphisms of Tournaments

By the Holland Representation Theorem, every lattice ordered group (l-group) is isomorphic to a subalgebra of the l-group of automorphisms of a chain. Since weakly associative lattice groups (wal-groups) and tournaments are non-transitive generalizations of l-groups and chains, respectively, the problem concerning the possibility of representation of wal-groups by automorphisms of tournaments arises. In the paper we describe the class of wal-groups isomorphic to wal-groups of automorphisms of tournament and show some of its properties.     

A generalization of 2-Baer groups

A group in which all cyclic subgroups are 2-subnormal is called a 2-Baer group. The topic of this paper are generalized 2-Baer groups, i.e., groups in which the non-2-subnormal cyclic subgroups generate a proper subgroup of the group. If this subgroup is non-trivial, the group is called a generalized T₂-group. In particular, we provide structure results for such groups, investigate their nilpotency class, and construct examples of finite p-groups which are generalized T₂-groups.     

On Hu-subgroups and anti fuzzy Hu-subgroups

In this paper we define the concept of anti fuzzyH _v-subgroup of anH _v-group, and prove a few theorems concerning this concept. We also obtain a necessary and sufficient condition for a fuzzy subset of anH _v-group to be an anti fuzzyH _v-subgroup. We also abtain a relation between the fuzzyH _v-subgroups and the anti fuzzyH _v-subgroups.     

2-Groups with normal noncyclic subgroups

We study nearly-hamiltonian 2-groups (¯H₂-groups). A nonabelian 2-group having at least one proper noncyclic subgroup is called an ¯H₂-group if all noncyclic subgroups are normal.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 75–84, July, 1968.The author expresses his deep gratitude towards Professor S. N. Chernikov for guidance while writing this paper.     

Groupable lattices

A lattice is called groupable provided it can be endowed with the structure of an l-group (lattice ordered group). The primary objective of this paper is to introduce an order theoretic property of groupable lattices which implies that all associated l-groups are subdirect products of totally ordered groups. This is an analog to Iwasawa's well-known result which asserts that a conditionally complete l-group is abelian. A secondary objective is to outline a general method for identifying classes of l-groups determined by order theoretic properties.     

On algorithms to compute some H V-groups

In this paper, we consider hyperstructures (H, ·) whenH = {e, a, b}. We put a condition on (H, ·) wheree is a unit. We obtain minimal and maximalH _v-groups, semigroups and quasigroups, using Mathematica 3.0 computer programs.     

Finite groups with ℱ-subnormal conditions

Let ? be a subgroup-closed saturated formation. A finite group G is called an ?_pc-group provided that each subgroup X of G is ?-subabnormal in the ?-subnormal closure of X in G. Let ?_pc be the class of all ?_pc-groups. We study some properties of ? _pc-groups and describe the structure of ?_pc-groups when ? is the class of all soluble π-closed groups, where π is a given nonempty set of prime numbers.     

Groups whose non-linear irreducible characters are rational valued

A finite group G all of whose nonlinear irreducible characters are rational is called a \mathbbQ₁{\mathbb{Q}_1}-group. In this paper, we obtain some results concerning the structure of \mathbbQ₁{\mathbb{Q}_1}-groups.     

Normal subgroups and elementary theories of lattice-ordered groups

We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H _i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H _i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S _i ) of infinite linearly ordered sets (S _i , ) such that each group A(S _i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.     

Minimal Non-FO-Groups

A group G is said to be a “minimal non-FO-group” (an MNFO-group) if all its proper subgroups are FO-groups, but G itself is not. The aim of this article is to study the class of MNFO-groups. The structure of MNFO-groups is completely described, both in nonperfect case and perfect case.     

Minimal Number of Generators and Minimum Order of a Non-Abelian Group Whose Elements Commute with Their Endomorphic Images

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p ⁸ for any odd prime number p and this order is 2⁷ for p = 2. Some of these results are proved for a class wider than the class of E-groups.