Powerfully nilpotent groups of maximal powerful class

In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime $$p>r$$, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.     

On odd order nilpotent groups with class 2

Let G be an odd order nilpotent group with class 2 and let e denote the exponent of its commutator subgroup. Let ${e=p_1^{r_1}p_2^{r_2}\ldots p_s^{r_s}}Let G be an odd order nilpotent group with class 2 and let e denote the exponent of its commutator subgroup. Let e=p₁^r₁p₂^r₂?p_s^r_s{e=p_1^{r_1}p_2^{r_2}\ldots p_s^{r_s}}, where the p _i ’s are the prime divisors of the order of G and the r _i ’s are non-negative integers. Then there are at least (?_i=1^s(1+r_i))-1{\left(\prod_{i=1}^{s}(1+r_i)\right)-1} non-isomorphic nilpotent groups with class 2 and each of the groups has the same order structure as G.     

Decompositions of finitely generated nilpotent groups of class 2

In this paper we prove that rational indecomposability is a genus property for finitely generated torsion-free nilpotent groups of class 2. We use this result to determine the genus of finitely generated torsion-free nilpotent groups of class 2 which decompose as a direct product of rationally indecomposable groups. Received: 3 November 2005     

Automorphism groups of nilpotent groups

We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .     

A counter example for nilpotent groups

Let R be a ring with 1, I be a nilpotent subring of R (there exists a natural number n, such that In = 0), and I be generated by {xj |j ∈ J} as ring. Write U = 1 + I, and it is a nilpotent group with class ≤ n - 1. Let G be the subgroup of U which is generated by {1 + xj|j ∈ J}. The group constructed in this paper indicates that the nilpotency class of G can be less than that of U.     

A class of nilpotent lie algebras

A p-filiform Lie algebra g is a nilpotent Lie algebra for which Goze’s invariant is (n–p,1,…,1). These Lie algebras are well known for P ≥ n-4n = dim(g). In this paper we describe the p-filiform Lie algebras, for p = n-5 and we gjive their classification when the derived subalgebra is maximal.     

Extended genus of torsion-free finitely generated nilpotent groups of class two

The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M _p , N _p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Compact nilpotent groups

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π₂X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T _π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T _π-compact provided G/T _πG is π-complete, extending the abelian group result in a natural way.