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.     

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.     

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.     

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.     

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.).     

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

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.     

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.     

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.     

On Units of Group Algebras of 2-Groups of Maximal Class

Abstract

We determine the number of elements of order two in the group of normalized units V(𝔽₂ G) of the group algebra 𝔽₂ G of a 2-group of maximal class over the field 𝔽₂ of two elements. As a consequence for the 2-groups G and H of maximal class we have that V(𝔽₂ G) and V(𝔽₂ H) are isomorphic if and only if G and H are isomorphic.     

子群的核平凡或正规闭包极大的有限p群

称有限 p 群 G 为ACT 群,如果对每个交换子群H, 其正规核 H_G=1 或 H_G=H. 又称p 群 G是CC 群,如果对每个非正规交换子群H, 有 H_G=1 或 H^G 在G中的指数为 p. 本文分类了ACT 群和CC 群.     

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.      

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerp ⁿ, theH _p ⁿ -subgroupH _pn (G) is defined byH _p ⁿ (G)=〈xεG|x ^pn≠1〉. A groupH≠1 is called aH _p ⁿ -group, if there is a finite groupG such thatH is isomorphic toH _p ⁿ (G) andH _p ⁿ (G)≠G. It is known that the Fitting length of a solvableH _p ⁿ -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH _p ⁿ -groups having Fitting lengthn, this result is “almost” best possible.     

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

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.     

Soluble minimal non-(finite-by-Baer)-groups

Let \mathfrakX{\mathfrak{X}} be a class of groups. A group G is called a minimal non- \mathfrakX{\mathfrak{X}}-group if it is not an \mathfrakX{\mathfrak{X}}-group but all of whose proper subgroups are \mathfrakX{\mathfrak{X}}-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G ^′′ is a minimal non-nilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ _n (G′) is a minimal non-(finite-by-abelian)-group.     

On a class of groups of prime-power order

LetG be a finitep-group,d(G)=dimH ¹ (G, Z _p) andr(G)=dimH ²(G, Z_p). Thend(G) is the minimal number of generators ofG, and we say thatG is a member of a classG _p of finitep-groups ifG has a presentation withd(G) generators andr(G) relations. We show that ifG is any finitep-group, thenG is the direct factor of a member ofG _p by a member ofG _p .     

Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic

We study the subgroup structure of some two-generator p-groups and apply the obtained results to metacyclic p-groups. For metacyclic p-groups G, p > 2, we do the following: (a) compute the number of nonabelian subgroups with given derived subgroup, show that (ii) minimal nonabelian subgroups have equal order, (c) maximal abelian subgroups have equal order, (d) every maximal abelian subgroup is contained in a minimal nonabelian subgroup and all maximal subgroups of any minimal nonabelian subgroup are maximal abelian in G. We prove the same results for metacyclic 2-groups (e) with abelian subgroup of index p, (f) without epimorphic image ? D₈. The metacyclic p-groups containing (g) a minimal nonabelian subgroup of order p ⁴, (h) a maximal abelian subgroup of order p ³ are classified. We also classify the metacyclic p-groups, p > 2, all of whose minimal nonabelian subgroups have equal exponent. It appears that, with few exceptions, a metacyclic p-group has a chief series all of whose members are characteristic.     

Topologies arising from metrics valued in abelian <Emphasis Type="Italic">ℓ</Emphasis>-groups

This paper considers metrics valued in abelian ℓ-groups and their induced topologies. In addition to a metric into an ℓ-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special case, we discuss a topology on a lattice ordered abelian group from the metric d _G and the positive filter consisting of the weak units of G; in the case of \mathbb Rⁿ{\mathbb R^{n}} , this is the Euclidean topology. We also show that there are many Nachbin convex topologies on an ℓ-group which are not induced by any positive filter of the ℓ-group.