A Mal’tsev basis for a partially commutative nilpotent metabelian group

Let be a group of prime exponent . In this paper we prove that is nilpotent of class at most 2 if and only if one of the following properties is true: is also the support of a commutative group such that and have the same cyclic cosets [cosets of order ]. the operation defined on by putting , gives a structure of commutative group.\end

Some finitely presented nilpotent-by-abelian groups

A group in which every element commutes with its endomorphic images is called an “E-group″. If p is a prime number, a p-group G which is an E-group is called a “pE-group″. Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p 8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups.

In this paper, we give a generalization of the concept of group commutativity. A group [Formula: see text] will be defined as [Formula: see text]-abelian (or [Formula: see text]-commutative), where [Formula: see text] is the alternating group of degree [Formula: see text], if [Formula: see text], for every [Formula: see text], and every pairwise non-commuting elements [Formula: see text]. We prove that [Formula: see text]-commutativity implies [Formula: see text]-commutativity for every [Formula: see text]. For [Formula: see text]-groups, we prove that the converse holds. We provide some sufficient condition for [Formula: see text]-commutativity, and then we prove that Hamiltonian groups and minimal non-abelian [Formula: see text]-groups are [Formula: see text]-abelian. For finite groups, we prove that a non-abelian and [Formula: see text]-abelian group is nilpotent of class two and has even order. A partial description of such groups is given.

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory.

It is shown that, if Q is a finitely generated abelian group, a finitely generated Q-group A is m-tame if and only if the mth tensor power of the augmentation ideal of ZA is finitely generated over Am ⋊ Q, where Q acts diagonally on both Am and the tensor power. It is proved that quotients of metabelian groups of type FP3 are again of type FP3, and a necessary condition is found for a split extension of abelian-by-(nilpotent of class two) groups to be of type FP2. A conjecture is formulated that generalises the FPm-Conjecture for metabelian groups, and it is shown that one of the implications holds in the prime characteristic case.

The study of homology groups ${\mathfrak {B}_n}(\Pi ,A),\mathfrak {B}$ a variety, $\Pi$ a group in $\mathfrak {B}$, and $A$ a suitable $\Pi$-module, is continued. A ’Tor’ is constructed which gives a better (but imperfect) approximation to these groups than a Tor previously considered. ${\mathfrak {B}_2}(\Pi ,Z)$ is calculated for various varieties $\mathfrak {B}$ and groups $\Pi$.

Groupes d'Automorphismes des Boucles de Moufang Commutatives

It is natural to describe the structure of an arbitrary finite group G in terms of the length of these or those normal series with ‘nice’ factors. Such length-type parameters are well-known for soluble finite groups: the derived length, the nilpotency class, the Fitting height (nilpotent length). An important parameter is also the p-length of a p-soluble group.

Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

In this paper, we study the capacity/entropy region of finite, directed, acyclic, multiple-sources, and multiple-sinks network by means of group theory and entropy vectors coming from groups. There is a one-to-one correspondence between the entropy vector of a collection of \(n\) random variables and a certain group-characterizable vector obtained from a finite group and \(n\) of its subgroups. We are looking at nilpotent group characterizable entropy vectors and show that they are all also abelian group characterizable, and hence they satisfy the Ingleton inequality. It is known that not all entropic vectors can be obtained from abelian groups, so our result implies that to get more exotic entropic vectors, one has to go at least to soluble groups or larger nilpotency classes. The result also implies that Ingleton inequality is satisfied by nilpotent groups of bounded class, depending on the order of the group.

In this chapter we shall have a look at what sort of group has a faithful representation of finite degree over a field. Our plan of campaign is roughly as follows. Firstly we consider abelian groups, and then soluble groups —especially those satisfying either the minimal condition or the maximal condition on subgroups. Next we prove a local theorem of Mal’cev, which states that a group has a faithful representation of degree n if and only if each of its finitely generated subgroups has such a representation. Then we prove that a free group has a faithful representation of degree 2 over most fields, and discuss similar results for certain relatively free groups. We follow this with Nisnevic’s Theorem, that a free product of groups having faithful representations of degree n over fields of characteristic p≥0, has a faithful representation of degree n +1 over some field of characteristic p, plus some related results. Finally we consider the representability of wreath products of linear groups.

In this paper, be the non-abelian metabelian group and denotes the automorphism group of group . A group is metabelian group if its commutator group is abelian or if it has an abelian normal subgroupin which the factor group is also abelian. There are 35 non-Abelian metabelian groups of order less than or equal to 24. In this study, we investigate the automorphism groups of all non-abelian metabelian groups of order less than or equal to 24 and the verification has been made through GAP (Groups Algorithm Programming) software.

Let G≅C n 1 ⊕...⊕C n r be a finite and nontrivial abelian group with n 1 |n 2 |...|n r . A conjecture of Hamidoune says that if W=w 1 ·...·w n is a sequence of integers, all but at most one relatively prime to |G|, and S is a sequence over G with |S|≥|W|+|G|-1≥|G|+1, the maximum multiplicity of S at most |W|, and σ(W)≡0mod|G|, then there exists a nontrivial subgroup H such that every element g∈H can be represented as a weighted subsequence sum of the form g=∑ n i=1w i s i , with s 1 ·...·s n a subsequence of S. We give two examples showing this does not hold in general, and characterize the counterexamples for large |W|≥1 2|G|.A theorem of Gao, generalizing an older result of Olson, says that if G is a finite abelian group, and S is a sequence over G with |S|≥|G|+𝔻(G)-1, then either every element of G can be represented as a |G|-term subsequence sum from S, or there exists a coset g+H such that all but at most |G/H|-2 terms of S are from g+H. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis |S|≥|G|+𝔻(G)-1 can be relaxed to |S|≥|G|+d * (G), where d * (G)=∑ r i=1(n i -1). We also use this method to derive a variation on Hamidoune’s conjecture valid when at least d * (G) of the w i are relatively prime to |G|.