COMPLETIONS AND CATEGORICAL COMPACTNESS FOR NILPOTENT GROUPS

Abstract

We consider reflection functors in the category of nilpotent groups satisfying certain exactness properties for which the Mal'cev completion functor and the p-cotorsion completion functors are prototypical examples. Each of these functors defines a generalized torsion theory, which in turn defines a closure operator on subgroups. This gives rise to the notion of a categorically compact group with respect to the closure operator which we characterize. This approach provides a unified treatment for the categorically compact groups with respect to the Mal'cev completion and with respect to the p-cotorsion completion, the latter being new. We also consider the p-pro-finite completion, suitably restricted to obtain a reflection functor, and characterize the compact groups so arising.     

NILPOTENT GROUPS AND ABELIANIZATION

We study the question of what properties of nilpotent groups are shared by their abelianizations. We identify two such properties—that of being a π-torsion group, where π is a family of primes, and that of having q^th roots, for some prime q. We use these properties to provide simplified proofs of the following theorems in the localization of nilpotent groups.

Let H, K be subgroups of the nilpotent group N and let P be a family of primes. Then [H, K] P = [H_P, K_p]

Let the group G act on the nilpotent group N. Then G acts compatibly on N_p and (Γ^G _i N)_P = Γ^G _i(N_p).

The second theorem above is then applied to the study of the localization of relative groups, in the sense of [4].     

QUASIREGULAR TORSION RINGS HAVING ISOMORPHIC ADDITIVE AND CIRCLE COMPOSITION GROUPS

Abstract

We investigate the role played by torsion properties in determining whether or not a commutative quasiregular ring has its additive and circle composition (or adjoint) groups isomorphic. We clarify and extend some results for nil rings, showing, in particular, that an arbitrary torsion nil ring has the isomorphic groups property if and only if the components from its primary decomposition into p-rings do too.

We look at the more specific case of finite rings, extending the work of others to show that a non-trivial ring with the isomorphic groups property can be constructed if the additive group has one of the following groups in its decomposition into cyclic groups: Z₂ ⁿ (for n ≥ 3), Z₂ ⊕ Z₂ ⊕ Z₂, Z₂ ⊕ Z₄, Z₄ ⊕ Z₄, Z _p ⊕ Z _p (for odd primes, p), or Z _p ⁿ (for odd primes, p, and n ≥ 2).

We consider, also, an example of a ring constructed on an infinite torsion group and use a specific case of this to show that the isomorphic groups property is not hereditary.     

ON NONABELIAN TENSOR PRODUCT MODULO q OF GROUPS

The nonabelian tensor product modulo q of two crossed modules of groups is investigated, where q is a positive integer. It is obtained a six term exact sequence of groups connecting the nonabelian tensor product modulo q with algebraic K-functor K ₂ with Z _q coefficients for (noncommutative) local rings. The notion of q-homology groups of a group G with coefficients in a G-module A is introduced, some its properties and calculations are given. The relationship between q-homology groups and derived functors of tensor product modulo q is studied.     

ON PARTIALLY SATURATED FORMATIONS AND RESIDUALS OF FINITE GROUPS

Partially saturated formations of finite groups are studied with the help of group functions. The results are applied for studying F-groups of automorphisms and π-complements to F-residuals.     

On chief factors of finite groups

Let H and K be normal subgroups of a finite group G and let K≤H. If A is a subgroup of G such that AH=AK or A∩H=A∩K, we say that A covers or avoids H/K respectively. The purpose of this paper is to investigate factor groups of a finite group G using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results.     

ON ADDITIVE GROUPS OF SEMI-LOCAL RINGS

Abstract

Results are obtained on the additive groups of semi-local rings. Conditions are given for embedding a ring into a local ring.     

CATEGORIES OF TOPOLOGICAL GROUPS

Abstract

This paper is a survey of recent (and some not so recent, results concerning categorical constructions on topological groups, with particular emphasis on free topological groups and coproducts (free products) of topological groups. An extensive bibliography is included.     

A MAXIMAL CLASS OF ABELIAN GROUPS ARISING FROM THE ISOMORPHISM G = T[EXT[G.X)]

Abstract

We characterize the class of groups G having the property that G ? T[Ext(G,X)] for some group X. We show that for every prime p, the p-component of such a group G has the form T [π∞_n=1 Z(pn)^mpn] where m_pn is finite for every prime p and natural number n.     

The influence of $\pi$-quasinormality of some subgroups of a finite group

A subgroup H of G is said to be $\pi$-quasinormal in G if it permute with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of G are $\pi$-quasinormal in G. The main result we proved in this paper is the following:Theorem 3.4. Let ${\cal F}$ be a saturated formation containing the supersolvable groups. Suppose that G is a group with a normal subgroup H such that $G/H \in {\cal F}$, and all maximal subgroups of any Sylow subgroup of $F^{*}(H)$ are $\pi$-quasinormal in G, then $G \in {\cal F}$. Received: 10 May 2002     

Groups with All Subgroups Pronormal-by-Finite

A subgroup X of a group G is called pronormal-by-finite if there exists a pronormal subgroup Y of G such that Y ≤ X and |X : Y| is finite. The structure of (generalized) soluble groups in which all subgroups are pronormal-by-finite is investigated. Among other results, it is proved in particular that a finitely generated soluble group with such property is central-by-finite, provided that it has no infinite dihedral sections.     

FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS

Abstract

This paper investigates torsion-free abelian groups A which are Q E-flat, i.e. for which Q A is flat as an Q E(A)-module. It is shown that a torsion-free A has this property iff Tor¹ (M, A) is torsion for all right E(A)-modules M. Furthermore, a torsion-free group of rank 4 is constructed which is Q E-flat but not quasi-isomorphic to an E-flat group. This gives a negative response to a question of R. Pierce. The paper concludes with a discussion of the structure of torsion-free groups of finite rank which are Q E-flat.     

TORSION-FREE ABELIAN GROUPS WITH FINITE RANK ENDOMORPHISM RINGS

Abstract

We show that if A and C are torsion-free abelian groups with ∩{ker f|f: A → C} = 0 = ∩{ker g|g: C → A}, and if A has a left Artinian quasi-endomorphism ring then A and C share a nonzero quasi-summand. Some consequences explored.     

WHEN ARE DIVISIBLE ABELIAN GROUPS INJECTIVE?

Abstract

The above question is considered, in the categories ShL of sheaves on a local lattice L and MEns of sets acted upon by a monoid M, for either all divisible abelian groups or all torsionfree divisible abelian groups, the aim being to characterize those L and M for which these types of abelian groups are Injective. Typical results: All divisible abelian groups are injective (i) in ShL iff L is Boolean, (ii) in MEns, M left or right cancellative, iff M is trivial, and (iii) in MEns, M commutative iff M is finite and idempotent.     

The <Emphasis Type="Italic">K</Emphasis>(π,1)-conjecture for the affine braid groups

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on ⁿ is known to be a K(,1) space for the corresponding Artin group $\Cal A$. A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type à _n and its associated Artin group, the affine braid group $\tilde{\Cal A}$. Using the fact that $\tilde{\Cal A}$ can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, $n$-dimensional K(,1)-space for $\tilde{\Cal A}$, and use it to prove the K(,1) conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.     

On the composition length of finite primitive linear groups

LetG be a finite primitive linear group over a fieldK, whereK is a finite field or a number field. We bound the composition length ofG in terms of the dimension of the underlying vector space and of the degree ofK over its prime subfield. As a byproduct, we prove a result of number theory which bounds the number of prime factors (counting multiplicities) ofq ⁿ−1, whereq, n>1 are integers, improving a result of Turull and Zame [6].     

ON THE STRUCTURE OF MINIMAL R n ACTIONS

Abstract

In this paper we study several concepts and models which are relevant in describing both the topological and dynamical structure of a typical R ⁿ flow. Some of these ideas originated in our earlier papers, and those of other authors, and we here attempt to synthesise these concepts. We start with shear—a notion which describes how little equicontinuity the flow contains. We move to R ⁿ suspensions which depend on particular R ⁿ cocycles and easily obtain a crude representation of the flow as a tower—a partial suspension over a base flow which contains the shear. Rudolph's deep theory of suspension models is modified to provide a new suspension model which incorporates the shear as the base of the tower. Finally we investigate towers in the context of a special class of automorphisms to see when these objects are themselves suspensions.     

THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS

Abstract

It is shown that Aut ?, the group of homeomorphisms of the rational numbers with the usual topology, has 2 ^N_o orbits on the power set P(?). We call S ? ? a moiety if S and its complement in ? are infinite. It is shown that the orbit of any moiety S under Aut ? has cardinality 2^N_o while the orbit of S under Aut(?, ≤), the group of order preserving automorphisms of ?, has cardinality N_o if and only if S is a finite union of disjoint rational intervals with rational endpoints.     

NILPOTENCY,SOLVABILITY AND RADICALS IN CATEGORIES

Abstract

Nilpotent and solvable ideals are defined and investigated in categories. The relation between the prime radical and the sum of the solvable ideals (which is also a radical) is discussed in categories. For example: If an object satisfies the maximal condition for ideals, then the prime radical is equal to the sum of the solvable ideals. Certain generalizations of theorems in rings, groups, Lie algebras, etc. are also proven, for example: An ideal α: I → A is semiprime if and only if A/I contains no non-zero nilpotent ideals.     

ON QUOTIENTS OF BRAID GROUPS

ABSTRACT.

Let G be the group ?[t, t ^?1] x ?. By studying the action of the braid group B_n on the set Gⁿ , we obtain representations of B_n into a wreath product of the symmetric group and the general linear group over ?[t, t ^?1]. This in particular recovers the Burau representation of the braid group. Furthermore, some quotients of the braid group are obtained by using the representations found.