Soluble Groups with Extremal Conditions on Commutators (original) (raw)

On soluble groups whose subnormal subgroups are inert

International Journal of Group Theory, 2015

A subgroup H of a group G is called inert if, for each g ∈ G, the index of H ∩ H g in H is finite. We give a classification of soluble-by-finite groups G in which subnormal subgroups are inert in the cases where G has no nontrivial torsion normal subgroups or G is finitely generated.

Locally soluble groups with min-n

Journal of the Australian Mathematical Society, 1974

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of c...

A question in the theory of saturated formations of finite soluble groups

Israel Journal of Mathematics, 1993

This paper examines the following question. If 7"/ and ~r are saturated formations then 7~-is defined to be the class of all soluble groups whose ~-normalizers belong to ~r. In general 7-/~r is a formation, but need not be a saturated formation. Here the smallest saturated formation containing ~/~r is studied. Introduction. Preliminaries All groups considered in this paper are finite and soluble. The reader is assumed to be familiar with the theory of saturated formations of finite soluble groups. We shall adhere to the notation used in [6]: this book is the main reference for the basic notation, terminology and results.

On trifactorized soluble minimax groups

Archiv der Mathematik, 1988

1. Introduction. In Kegel considered finite groups G = A B = A C = B C which are the product of three subgroups A, B, C where A and B are nilpotent. He showed that G is nilpotent or supersoluble, if C is nilpotent or supersoluble, respectively; see also Pennington [7]. In the following we extend these results to soluble-by-finite minimax groups. Recall that a soluble-by-finite group G is a minimax group if it has a series of finite length whose factors are finite or infinite cyclic or quasicyclic of type p~. The number m (G) of infinite factors in such a series is called the minimax rank of G.

On infinitely presented soluble groups

2010

We exhibit an infinitely presented 4-soluble group with Cantor-Bendixson rank one, and consequently with no minimal presentation. Then we study the class of infinitely presented metabelian groups lying in the condensation part of the space of marked groups.

Group algebras of some generalised soluble groups

Proceedings of the Edinburgh Mathematical Society, 1972

1. In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups.