Minimal Condition For Normal Subgroups Research Articles

Some classes of groups with weak minimal condition for normal subgroups

Some classes of groups with weak minimal condition for normal subgroups

Two-step solvable groups with weak minimal condition for normal subgroups

Two-step solvable groups with weak minimal condition for normal subgroups

Representations of metabelian groups satisfying the minimal condition for normal subgroups

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.

A note on 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.

On the construction of soluble groups satisfying the minimal condition for normal subgroups

A general method is described for constructing examples of soluble groups whose normal subgroups form a well-ordered chain under the ordering of inclusion. This method is a variant of one introduced in a recent paper by Heineken and Wilson. Each of the resulting groups is obtained by an embedding procedure from a pair of iterated wreath products A1 wr A2 wr … wr An, B1 wr B2 wr … wr Bn, where the constituent groups Ai., Bi are each either cyclic of prime power order or quasicyclic. Here n may be chosen arbitrarily, and the choice of constituent groups is subject only to a condition on the sequences of prime numbers that may occur as orders of elements in the groupsrespectively. The construction is applied to give certain examples which illustrate the limitations of results on particular classes of soluble groups satisfying the minimal condition for normal subgroups obtained in recent papers by Hartley, McDougall, and the present author.

Metanilpotent groups satisfying the minimal condition for normal subgroups

Metanilpotent groups satisfying the minimal condition for normal subgroups

Locally soluble groups with min-n.

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.

Soluble groups satisfying the minimal condition for normal subgroups

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

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.

A characteristically simple group

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.