Maximal Condition For Normal Subgroups Research Articles

Palindromic Width of Finitely Generated Solvable Groups

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step ≥3, we prove that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of G is finite. We also prove that the palindromic width of ℤ≀ℤ with respect to the set of standard generators is 3.

COMMUTATOR LENGTH OF SOLVABLE GROUPS SATISFYING MAX-N

In this paper we find a suitable bound for the number of commutators which is required to express every element of the derived group of a solvable group satisfying the maximal condition for normal subgroups. The precise formulas for expressing every element of the derived group to the minimal number of commutators are given.

Some examples of finiteness conditions in centre-by-metabelian groups

AbstractCentre-by-metabelian groups with the maximal condition for normal subgroups are exhibited which (a) are residually finite but have quotient groups which are not residually finite; and (b) have all quotients residually finite but are not abelian-by-polycyclic.

Finitely presented solvable groups that do not satisfy the maximal condition for normal subgroups

Finitely presented solvable groups that do not satisfy the maximal condition for normal subgroups

The number of indecomposable terms in direct decompositions of groups with the maximal condition

If nt and n are positive integers, m > 1, n > 1, then there exists a finitely generated torsion free nilpotent group which can be expressed as a direct product of n directly indecomposable groups and as a direct product of m directly indecomposable groups such that no factor in the first decomposition is isomorphic to a factor in the second [ 11. This leads us to ask: If a group G satisfies the maximal condition for normal subgroups, what conditions will guarantee that any two direct decompositions of G into nontrivially directly indecomposable factors have the same number of factors? If G has the above property we call G a UNIF group ( a group with a unique number of indecomposable factors in direct products). The object of this paper is to study the UNIF property. In Section 3 we show that every group G with the maximal condition for normal subgroups which fails to have the UNIF property gives rise to a directly indecomposable homomorphic image M such that for some finitely generated free abelian group F, M x F is not a UNIF group. Section 4 contains a refinement of this result. In Section 5 we use the results of Sections 3 and 4 to show that a group G which obeys the maximal condition for normal subgroups has the UNIF property if either of the following holds:

Direct decompositions of groups with finitely generated commutator quotient group

AbstractLet G/G' be finitely generated and let G = B1 x A1 = B2 x A2 = … = Bi x Ai = … with each Bi isomorphic to a fixed group B which obeys the maximal condition for normal subgroups. Then the Ai represent only finitely many isomorphism classes. We give an example with B infinite cyclic, G/G' free abelian of infinite (countable) rank and such that G is decomposed as above with no two Ai isomorphic.

The equivalence of ×^{𝑡}𝐶≈×^{𝑡}𝐷 and 𝐽×𝐶≈𝐽×𝐷

Let C satisfy the maximal condition for normal subgroups and let × t C ≈ × t D \times {\,^t}C\, \approx \, \times {\,^t}D for some positive integer t. Then C × J ≈ D × J C\, \times \,J\, \approx \,D\, \times \,J where J is the infinite cyclic group. If × s C ≈ × t D \times {\,^s}C\, \approx \, \times {\,^t}D and s ⩾ t s \geqslant \,t , there exists a finitely generated free abelian group S such that C is a direct factor of D × S D\, \times \,S .

Finitely Presented Centre-By-Metabelian Groups

An investigation of the groups described in the title is carried out. It is shown that all such groups are extensions of an abelian group by a polycyclic group and so have the maximal condition for normal subgroups and are residually finite.

Some cancellation theorems with applications to nilpotent groups

AbstractIf C is a group which satisfies the maximal condition for normal subgroups, then C may be cancelled from a group A in direct products if and only if the infinite cyclic group can be cancelled from A. Finitely generated torsion free nilpotent groups of class 2 satisfy a Remak Krull Schmidt condition.

On characteristically simple groups

1·1. A group is called characteristically simple if it has no proper non-trivial subgroups which are left invariant by all of its automorphisms. One familiar class of characteristically simple groups consists of all direct powers of simple groups: this contains all finite characteristically simple groups, and, more generally, all characteristically simple groups having minimal normal subgroups. However not all characteristically simple groups lie in this class because, for instance, additive groups of fields are characteristically simple. Our object here is to construct finitely generated groups, and also groups satisfying the maximal condition for normal subgroups, which are characteristically simple but which are not direct powers of simple groups.

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.

Relatively free, nearly nilpotent groups

We show that a free nilpotent group of countable rank, as well as a free group of countable rank of the variety defined by the identity [[x1 x2,..., xn], [xn+1, xn+2]]=1, satisfies the maximal condition for normal subgroups admitting endomorphisms induced by order preserving one-to-one mappings of the set of free generators into itself.

Groups with every proper quotient finite

1·1. We shall write for the class of groups all of whose non-trivial normal subgroups have finite index. Thus, rather obviously, finite groups, simple groups, and the infinite cyclic and dihedral groups all lie in the class . Other examples of -groups are to be found in a variety of contexts. The main result of Mennicke (8) shows that, for n ≥ 3, the factor group of SL(n, Z) by its centre is a -group, where SL(n, Z) denotes the unimodular group of all n × n invertible matrices with integer coefficients and with determinant 1. In McLain(7), an example is given of an infinite, locally finite, locally soluble -group, whose only non-trivial normal subgroups are the terms of its derived series, and in (4), P. Hall discusses an infinite -group, all of whose proper quotients are finite p-groups. If is a class of groups closed under taking homomorphic images, it is easily seen that the existence of an infinite -group satisfying the maximal condition for normal subgroups implies the existence of an infinite -group in the class , so that certain questions concerning the finiteness of groups satisfying the maximal condition for normal subgroups can be interpreted as questions about -groups.

On R0-Closed Classes, and Finitely Generated Groups

1.1. If a group satisfies the maximal condition for normal subgroups, then all its central factors are necessarily finitely generated. In [2], Hall asked whether there exist finitely generated soluble groups which do not satisfy the maximal condition for normal subgroups but all of whose central factors are finitely generated. We shall answer this question in the affirmative. We shall also construct a finitely generated group all of whose subnormal subgroups are perfect (and which therefore has no non-trivial central factors), but which does not satisfy the maximal condition for normal subgroups. Related to these examples is the question of which classes of finitely generated groups satisfy the maximal condition for normal subgroups. A characterization of such classes has been obtained by Hall, and we shall include his result as our first theorem.

Cancellation of groups with maximal condition

It is not true that a group which obeys the maximal condition for normal subgroups may always be cancelled in direct products. However, we show the following Theorem. Let C C be a group which obeys the maximal condition for normal subgroup. Suppose further that if C ∗ {C_{\ast }} is an arbitrary homomorphic image of C C , then C ∗ {C_{\ast }} is not isomorphic to a proper normal subgroup of itself. Then C C may be cancelled in direct products. Some generalizations of this result are indicated.

Groups satisfying the maximal condition for normal subgroups

Groups satisfying the maximal condition for normal subgroups

On certain soluble groups

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?