On pure subnormal operators with finite rank self-commutators and related operator tuples

In this paper, we study the model of a pure subnormal operator with finite rank self-commutator and of the relatedn-tuple of commuting linear bounded operators. We also give some applications of the model to the theory ofn-tuples of commuting operators with trace class self-commutators.This work is supported in part by a NSF grant no. DMS-9400766.     

On the duals of subnormal tuples

A notion of the dual of a subnormal tuple of operators is discussed.     

Trace formulas for a class of subnormal tuples of operators

Trace formulas are established for the product of commutators related to subnormal tuple of operators (S ₁,...,S _n ) with minimal normal extension (N ₁,...,N _n ) satisfying conditions that sp(S _j )/sp(N _j ) is simply-connected with smooth boundary Jordan curve sp(N _i ) and [S _j ^* ,S _j ]^1/2 L ¹,j=1, 2,...,n.Some complete unitary invariants related to the trace formulas are found.This work is supported in part by NSF Grant no. DMS-9101268.     

A remark on the spectral multiplicity of normal extensions of commuting subnormal operator tuples

The authors gratefully acknowledge the support of the National Science Foundation.     

Conjugation of two finite subnormal subgroups

A necessary and sufficient condition for two finite subnormal subgroups to be conjugate is presented in this paper.     

Permutable subnormal subgroups of finite groups

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called -groups. In particular, it is shown that the finite solvable -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not -groups but all subnormal subgroups of defect two are permutable. Received: 13 August 2008     

Hypercentral groups of finite subnormal rank

We introduce the notion of subnormal rank of a group and study hypercentral groups of finite subnormal rank. We construct an example of a hypercentral group that has a finite subnormal rank and infinite (special) rank.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1577–1580, November, 1995.     

Comonolithic subnormal subgroups generating a finite group

Every finite group is generated by minimal sets of subnormal comonolithic subgroups all this sets having the same number of members and whose perfect subgroups being the same for every such a set. The proof of this result depends on some well known results about subnormal subgroups, mainly due to H. Wielandt.     

The product of generalized subnormal subgroups in finite groups

We study the structure of some classes of finite groups with a given system of commuting generalized subnormal subgroups. In particular, our results yield a characterization of superradical and Shemetkov formations.     

Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov

Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.     

Generating tuples of free products

Grushko's theorem [Mat. Sb. 8 (1940) 169–182] says thatany generating tuple (g₁, ..., g_m) of a free product H*K isNielsen-equivalent to a tuple (h₁, ..., h_l, k_l+1, ..., k_m) withh_i H and k_i K for all i. The h_i and k_i are clearly not unique.In this paper we address the extent of this non-uniqueness.     

Characterization of classes of finite groups with the use of generalized subnormal Sylow subgroups

A characterization of the classes of all π-nilpotent, all π-closed, and all π-decomposable finite groups is obtained by using generalized subnormal Sylow subgroups.     

Quadrics of finite type

We show that the only quadrics of finite type in Euclidean 3-space are the circular cylinder and the sphere.This work was done while the second author was visiting the Michigan State University. He would like to thank MSU, and in particular the staff of the department of mathematics for their hospitality.Research Assistant of the National Fund for Scientific Research (Belgium)