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.

scholarly journals Representations of metabelian groups satisfying the minimal condition for normal subgroups

1976 ◽  
Vol 14 (2) ◽  
pp. 267-278 ◽  
Author(s):  
Howard L. Silcock
Keyword(s):  
Minimal Condition ◽  
Normal Subgroups ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Derived Length ◽  
Indecomposable Representations

A question of John S. Wilson concerning indecomposable representations of metabelian groups satisfying the minimal condition for normal subgroups is answered negatively, by means of an example. It is shown that such representations need not be irreducible, even when the group being represented is an extension of an elementary abelian p–group by a quasicyclic q–group of the type first described by V.S. Čarin, and the characteristic of the field is a prime distinct from both p and q. This implies that certain techniques used in the study of metabelian groups satisfying the minimal condition for normal subgroups are not available for the corresponding class of soluble groups of derived length 3.

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

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.

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?

Groups Satisfying the Minimal Condition on Subnormal Non-normal Subgroups

2006 ◽  
Vol 13 (03) ◽  
pp. 411-420 ◽  
Author(s):  
Fausto De Mari ◽  
Francesco de Giovanni
Keyword(s):  
Minimal Condition ◽  
Normal Subgroups ◽  
Transitive Relation ◽  
Soluble Groups

In this paper, the structure of (generalized) soluble groups for which the set of all subnormal non-normal subgroups satisfies the minimal condition is described, taking as a model the known theory of groups in which normality is a transitive relation.

scholarly journals Some topological properties of residually Černikov groups

1982 ◽  
Vol 23 (1) ◽  
pp. 65-82 ◽  
Author(s):  
M. R. Dixon
Keyword(s):  
Finite Groups ◽  
Inverse Limit ◽  
Normal Subgroups ◽  
Topological Properties ◽  
Profinite Groups ◽  
Dense Subgroup ◽  
Locally Finite ◽  
Soluble Groups ◽  
Residually Finite ◽  
Residually Finite Group

In this paper we shall indicate how to generalise the concept of a cofinite group (see [7]). We recall that any residually finite group can be made into a topological group by taking as a basis of neighbourhoods of the identity precisely the normal subgroups of finite index. The class of compact cofinite groups is then easily seen to be the class of profinite groups, where a group is profinite if and only if it is an inverse limit of finite groups. It turns out that every cofinite group can be embedded as a dense subgroup of a profinite group. This has important consequences for the class of countable locally finite-soluble groups with finite Sylow p-subgroups for all primes p, as shown in [7] and [14].

scholarly journals A note on locally finite varieties

1976 ◽  
Vol 14 (1) ◽  
pp. 63-70 ◽  
Author(s):  
John S. Wilson
Keyword(s):  
Minimal Condition ◽  
Normal Subgroups ◽  
Maximal Condition ◽  
Locally Finite ◽  
Variety Of Groups ◽  
Locally Finite Varieties

Evidence is presented which suggests that the following assertions about a variety ⊻ of groups may be equivalent:(a) ⊻ is locally finite,(b) all ⊻-groups satisfying the maximal condition for normal subgroups are finite, and(c) all ⊻-groups satisfying the minimal condition for normal subgroups are finite.

A NOTE ON SOLUBLE GROUPS WITH THE MINIMAL CONDITION ON NORMAL SUBGROUPS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350134 ◽  
Author(s):  
M. DE FALCO ◽  
F. DE GIOVANNI ◽  
C. MUSELLA
Keyword(s):  
Finite Rank ◽  
Soluble Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Infinite Rank

It is known that there exist soluble groups of infinite rank which satisfy the minimal condition on normal subgroups. We prove here that if G is any soluble group satisfying the minimal condition on normal subgroups of infinite rank, then either G has finite rank or it satisfies the minimal condition on normal subgroups.

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