Luise-Charlotte Kappe

Rings in which derivations satisfy certain algebraic conditions

The primary purpose of this paper is to investigate some commutator conditions for rings, which were suggested by group-theoretic results of F, W. Levi, I. D. Macdonald and N. D. Gupta. Most of these conditions can be simply interpreted in terms of inner derivations, and they suggest further questions about arbitrary derivations. In [8] and [9] Levi essentially established the equivalence of the following properties for a group G: (i) The commutator operation in G is associative. (ii) Commutators are in the center of G. (iii) The commutator operation is right distributive with the group composition. (iv) The commutator operation is left distributive with the group composition. (Cf. Kurosh [7], pp. 99--101.) Denoting by [a, b] the group commutator alblab , we can reformulate conditions (i)--(iv) in terms of identities. The added condition (v) is obviously equivalent to (ii).

On commutators in p-groups

Abstract For each prime p, we determine the smallest integer n such that there exists a group of order pn in which the set of commutators does not form a subgroup. We show that n = 6 for any odd prime and n = 7 for p = 2.

Finite coverings by subgroups with a given property

Let be a group-theoretic property. We say a group has a finite covering by -subgroups if it is the set-theoretic union of finitely many -subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups, n -abelian subgroups or 2-central subgroups. R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [ G: Z C ( G )] finite implies the existence of a finite covering by subgroups of nilpotency class c , i.e. ℜ c -groups. However, an example of a group is given there which has a finite covering by ℜ 2 -groups, but Z 2 ( G ) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.

FINITE COVERINGS BY 2-ENGEL GROUPS

Baers characterisation of central-by-finite groups as groups possessing a finite covering by abelian subgroups is the starting point for this investigation. We characterise groups with a finite covering by 2-Engel subgroups as groups for which the subgroup of right 2-Engel elements has finite index; and the groups having a finite covering by normal 2Engel subgroups are exactly the 3-Eugel groups among those having a finite covering by 2-Engel subgroups. The second centre of a group having a finite covering by class two subgroups does not necessarily have finite index. However, a group has a finite covering by subgroups in a variety containing all cyclic groups if the margin of this variety in the group has finite index.

Infinite metacyclic groups and their non-abelian tensor squares

In this paper we classify all infinite metacyclic groups up to isomorphism and determine their non-abelian tensor squares. As an application we compute various other functors, among them are the exterior square, the symmetric product, and the second homology group for these groups. We show that an infinite non-abelian metacyclic group is capable if and only if it is isomorphic to the infinite dihedral group.

ON SUBGROUPS RELATED TO THE TENSOR CENTER

The tensor center of a group G is the set of elements a in G such that a ⊗ g = 1⊗ for all g in G.I t is ac haracteristic subgroup of G contained in its center. We introduce tensor analogues of various other subgroups of a group such as centralizers and 2-Engel elements and investigate their embedding in the group as well as interrelationships between those subgroups.

Subnormality conditions in non-torsion groups

According to results of Heineken and Stadelmann, a non-torsion group is a 2-Baer group if and only if it is 2-Engel, and it has all subgroups 2-subnormal if and only if it is nilpotent of class 2. We extend some of these results to values of n greater than 2. Any non-torsion group which is an n-Baer group is an n-Engel group. The converse holds for n = 3, and for all n in case of metabelian groups. A nontorsion group without involutions having all subgroups 3-subnormal has nilpotency class 4, and this bound is sharp.

On power margins

The characteristic subgroup M,,(G) has exponent n and will bc called the n-power margin of G. As an easy exercise it can be shown that M2(G) consists of the elements of order 2 in the center of G. This seems to be the only known fact about power margins beyong those which follow directly from the definition. Shedding some further light on power margins is the objective of this paper. In particular, we are interested in their embedding into the upper central series of a group, and the interrelationship between Engel elements and power margins. Since a group of exponent n coincides with its own n-power margin, it is reasonable to expect such embeddings only if all groups of exponent n are nilpotent of bounded class, namely if n = 2, 3 (cf. [S]). Meier-Wunderli’s theorem in [7], stating that a metabelian p-group of exponent p has nilpotency class at most p, suggests a result on the embedding of the p-power margin in the upper central series of a metabelian group. In addition, a sufficient condition will be given such that the @“-power margin is embedded in the hyperccnter of a group. Similar investigations have been done for n-Engel margins by T. Teague (cf. [lo]) and by the author (cf. [3]). For the variables X, y, x, we define the commutator word of weight 2 as [x, y] = .Y‘y-‘.xy; the commutator 337 002 l-8693/89

An analogue for semigroups of a group problem of P. Erdös and B. H. Neumann

3.00

On Subsets and Subgroups Defined by Commutators and Some Related Questions

In response to a question posed by P. Erdos, B. H. Neumann showed that in a group with every subset of pairwise noncommuting elements finite there is a bound on the size of these sets. Recently, H. E. Bell, A. A. Klein and the first author showed that a similar result holds for rings. However in the case of semigroups, finiteness of subsets of pairwise noncommuting elements does not assure the existence of a bound for their size. The largest class of semigroups in which we found Neumanns result valid are cancellative semigroups.