The weak minimal condition on subgroups which fail to be close to normal subgroups

2020 ◽  
Vol 560 ◽  
pp. 371-382
Author(s):  
Ulderico Dardano ◽  
Fausto De Mari ◽  
Silvana Rinauro
Keyword(s):  
Minimal Condition ◽  
Normal Subgroups ◽  
Weak Minimal Condition

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.

Weak minimal condition for nonnilpotent subgroups

1984 ◽  
Vol 23 (4) ◽  
pp. 303-306 ◽  
Author(s):  
A. N. Ostylovskii
Keyword(s):  
Minimal Condition ◽  
Weak Minimal Condition

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 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.

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