Soluble groups which are products of minimax groups

Bertram Huppert zum 60. Geburtstag     

Groups with minimax factor groups

We consider groups in which every normal subgroup which is not minimax determines a minimax quotient group. If G is a group with this property then it is clear that either G contains an ascending chain of normal subgroups with minimax quotient groups or G contains a normal minimax subgroup H such that G/H does not contain any non-identity normal minimax subgroups. In particular, every proper factor group of G/H is minimax. In the present paper we study the first case.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 620–625, May, 1990.     

Torsion-free groups with all subgroups subnormal

It is proved that a torsion-free group with all subgroups subnormal is nilpotent.     

Finite groups with subgroups supersoluble or subnormal

The aim of this paper is to study the structure of finite groups whose non-subnormal subgroups lie in some subclasses of the class of finite supersoluble groups.     

Finite groups with formation subnormal primary subgroups

Given a hereditary saturated formation F containing all nilpotent groups, we describe finite soluble groups in which every primary subgroup is self-normalizing or F-subnormal.     

Blocks with trivial intersection defect groups

We show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups, or is Morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non-abelian simple groups. In particular we classify all blocks of automorphism groups of non-abelian simple groups whose defect groups are non-cyclic and intersect pairwise trivially. A consequence is that Donovans conjecture holds for blocks whose defect groups intersect pairwise trivially.in final form: 14 January 2003Mathematics Subject Classification (2000): 20C20This research was supported in part by the Marsden Fund of New Zealand via grant UOA 810.     

Finite groups with generalized subnormal embedding of Sylow subgroups

Given a set π of primes and a hereditary saturated formation F, we study the properties of the class of groups G for which the identity subgroup and all Sylow p-subgroups are F-subnormal (K-F-subnormal) in G for each p in π. We show that such a class is a hereditary saturated formation and find its maximal inner local screen. Some criteria are obtained for the membership of a group in a hereditary saturated formation in terms of its formation subnormal Sylow subgroups.     

On trifactorized soluble minimax groups

This research was done while the last two authors were visitors at the University of Mainz. They are grateful to the Department of Mathematics for its excellent hospitality.     

Finite groups with subnormal second or third maximal subgroups

In the paper, the precise structure of finite groups all of whose second (or all of whose third) maximal subgroups are subnormal is established.     

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.     

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