scholarly journals On varieties of soluble groups II

1972 ◽  
Vol 7 (3) ◽  
pp. 437-441 ◽  
Author(s):  
J.R.J. Groves
Keyword(s):  
Nilpotent Group ◽  
Metabelian Groups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Finite Exponent

It is shown that, in a variety which does not contain all metabelian groups and is contained in a product of (finitely many) varieties each of which is soluble or locally finite, every group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent.

scholarly journals Injective modules and soluble groups satisfying the minimal condition for normal subgroups

1971 ◽  
Vol 4 (1) ◽  
pp. 113-135 ◽  
Author(s):  
B. Hartley ◽  
D. McDougall
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Divisible Group ◽  
Minimal Condition ◽  
Injective Hull ◽  
Normal Subgroups ◽  
Metabelian Groups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Injective Modules

Let p be a prime and let Q be a centre-by-finite p′-group. It is shown that the ZQ-modules which satisfy the minimal condition on submodules and have p–groups as their underlying additive groups can be classified in terms of the irreducible ZpQ-modules. If such a ZQ-module V is indecomposable it is either the ZpQ-injective hull W of an irreducible ZpQ-module (viewed as a ZQ-module) or is the submodule W[pn] of such a W consisting of the elements ω ∈ W which satisfy pnw = 0.This classification is used to classify certain abelian-by-nilpotent groups which satisfy Min-n, the minimal condition on normal subgroups. Among the groups to which our classification applies are all quasi-radicable metabelian groups with Min-n, and all metabelian groups which satisfy Min-n and have abelian Sylow p-subgroups for all p.It is also shown that if Q is any countable locally finite p'-group and V is a ZQ-module whose additive group is a p-group, then V can be embedded in a ZQ-module whose additive group is a minimal divisible group containing that of V. Some applications of this result are given.

On Metanilpotent Varieties of Groups

1970 ◽  
Vol 22 (4) ◽  
pp. 875-877 ◽  
Author(s):  
Narain Gupta
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finite Exponent ◽  
Torsion Free

Let denote the variety of all groups which are extensions of a nilpotent-of-class-c group by a nilpotent-of-class-d group, and let denote the variety of all metabelian groups. The main result of this paper is the following theorem.THEOREM. Let be a subvariety of which does not contain . Then every -group is an extension of a group of finite exponent by a nilpotent group by a group of finite exponent. In particular, a finitely generated torsion-free -group is a nilpotent-by-finite group.This generalizes the main theorem of Ŝmel′kin [4], where the same result is proved for subvarieties of , where is the variety of abelian groups. See also Lewin and Lewin [2] for a related discussion.

scholarly journals On varieties of soluble groups

1971 ◽  
Vol 5 (1) ◽  
pp. 95-109 ◽  
Author(s):  
J.R.J. Groves
Keyword(s):  
Metabelian Groups ◽  
Soluble Groups ◽  
Variety Of Groups ◽  
Finite Exponent

We show that, under certain conditions, a soluble variety of groups which does not contain the variety of all metabelian groups is a finite exponent by nilpotent by finite exponent variety.

Wreath products and p–groups

1959 ◽  
Vol 55 (3) ◽  
pp. 224-231 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Nilpotent Group ◽  
Wreath Product ◽  
Soluble Group ◽  
Nilpotent Groups ◽  
Wreath Products ◽  
Soluble Groups ◽  
Finite Exponent

The wreath product is a useful method for constructing new soluble groups from given ones (cf. P. Hall (3)). Now although the wreath product of one soluble group by another is (obviously) always soluble, the corresponding result is no longer true for nilpotent groups. It is the object of § 3 of this note to determine precisely when the wreath product W of a non-trivial nilpotent group A by a non-trivial nilpotent group B is nilpotent; in fact I prove that W is nilpotent if and only if both A and B are (nilpotent) p–groups with A of finite exponent and B finite.

scholarly journals Schunck classes and projectors in a class of locally finite groups

1995 ◽  
Vol 38 (3) ◽  
pp. 511-522 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Maximal Subgroups ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Finite Case ◽  
Saturated Formations ◽  
Definition Of ◽  
Schunck Class

We introduce a definition of a Schunck class of periodic abelian-by-finite soluble groups using major subgroups in place of the maximal subgroups used in Finite groups. This allows us to develop the theory as in the finite case proving the existence and conjugacy of projectors. Saturated formations are examples of Schunck classes and we are also able to obtain an infinite version of Gaschütz Ω-subgroups.

On Certain Classes of Locally Soluble Groups

1962 ◽  
Vol 58 (2) ◽  
pp. 185-195
Author(s):  
J. E. Roseblade
Keyword(s):  
Finitely Generated ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Nilpotent

A group G is called locally soluble if every finitely generated subgroup of G is soluble. Terms like ‘locally nilpotent’ and ‘locally finite’ are defined similarly.

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

On certain soluble groups

1969 ◽  
Vol 66 (1) ◽  
pp. 1-4 ◽  
Author(s):  
C. K. Gupta
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Finitely Generated Groups ◽  
The Law

In (2), Hall considered the question: for what varieties of soluble groups do all finitely generated groups satisfy max-n (the maximal condition for normal subgroups)? He has shown that the variety M of metabelian groups and more generally the variety of Abelian-by-nilpotent-of-class-c (c ≥ 1) groups has this property; whereas on the contrary, there are finitely generated groups in the variety V of centre-by-metabelian groups (i.e. defined by the law [x, y; u, v; z]) which do not satisfy max-n. One naturally raises the question: for what subvarieties of V do all finitely generated groups satisfy max-n?

A note on the cohomology of metabelian groups

1985 ◽  
Vol 98 (3) ◽  
pp. 437-445 ◽  
Author(s):  
P. H. Kropholler
Keyword(s):  
Finite Rank ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.

Formation theory and groups of automorphisms of -groups

1977 ◽  
Vol 76 (4) ◽  
pp. 255-265 ◽  
Author(s):  
M. J. Tomkinson
Keyword(s):  
Finite Groups ◽  
Composition Series ◽  
Formation Theory ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Finite Groups ◽  
Group Of Automorphisms ◽  
Infinite Case ◽  
Groups Of Automorphisms ◽  
Automorphisms Of Groups

SynopsisFurther results from the theory of finite soluble groups are extended to the class of locally finite groups with a satisfactory Sylow structure. Let be a saturated U-formation and A a -group of automorphisms of the -group G. A is said to act -centrally on G if G has an A-composition series (Λσ/Vσ; σ ∈ ∑) such that A induces an f(p)-group of automorphisms in each p-factor Λσ/Vσ. We show that in this situation A is an -group, thus generalising the result of Schmid [8]. Associated results of Schmid and of Baer are also extended to the infinite case.

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