Groups with many normal-by-finite subgroups

A subgroup of a group is said to be normal-by-finite if the core of in has finite index in . In this article groups satisfying the minimal condition on subgroups which are not normal-by-finite and groups with finitely many conjugacy classes of subgroups which are not normal-by-finite are characterized.

     

Curves with infinitely many points of fixed degree

Thed-th symmetric productC ^(d) of a curveC defined over a fieldK is closely related to the set of points ofC of degree ≤d. IfK is a number field, then a conjecture of Lang [Hi] proved by Faltings [Fa2] implies ifC ^(d) (K) is an infinite set, then there is aK-rational covering ofC → ℙ _|K ¹ of degree ≤2d. As an application one gets that for fixed fieldK and fixedd there are only finitely many primes ι such that the set of all elliptic curves defined over some extensionsL ofK with [L∶K]≤d and withL-rational isogeny of degree ι is infinite.     

Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

Groups with the minimality condition for infinitely generated subgroups

We establish that, in the class of locally almost solvable groups, the minimality condition for infinitely generated subgroups is equivalent to the minimality condition for (all) subgroups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1280–1282, September, 1994.     

Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

Groups with finitely many normalizers of non-periodic subgroups

A theorem of Polovickiĭ states that any group with finitely many normalizers of subgroups is finite over its centre. Here we prove that the centre of a non-periodic group G has finite index if and only if G has finitely many normalizers of non-periodic subgroups.     

Groups with finitely many normalizers of non-abelian subgroups

Abstract A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite commutator subgroup. Here the structure of locally graded groups with finitely many normalizers of (infinite) non-abelian subgroups is investigated, and the above result is extended to this more general situation. Keywords: normalizer subgroup, metahamiltonian group Mathematics Subject Classification (2000): 20F24     

Groups with many subgroups having a transitive normality relation

A group is said to be aT-group if all its subnormal subgroups are normal. The structure of groups satisfying the minimal condition on subgroups that do not have the propertyT is investigated. Moreover, locally soluble groups with finitely many conjugacy classes of subgroups which are notT-groups are characterized.     

Groups with many subgroups having polycyclic-by-finite layers or derived subgroup

A group is called (PF)L if the subgroups generated by its elements having same order (finite or infinite) are polycyclic-by-finite. In the present paper we prove that a group is locally graded minimal non-((PF)L∪(𝔓𝔉)𝔄) if, and only if, it is non-perfect minimal non-FC, where (𝔓𝔉)𝔄 denotes the class of (polycyclic-by-finite)-by-abelian groups. We prove also that a group of infinite rank whose proper subgroups of infinite rank are in ((PF)L∪(𝔓𝔉)𝔄) is itself in ((PF)L∪(𝔓𝔉)𝔄) provided that it is locally (soluble-by-finite) without simple homomorphic images of infinite rank. Our last result concerns groups that satisfy the minimal condition on non-((PF)L∪(𝔓𝔉)𝔄)-subgroups.     

Toda flows with infinitely many variables

The Toda flow and related flows extend naturally to operators in Hilbert space and the purpose of this paper is to describe these flows and to analyse some of their special properties.     

Integrodifferential systems with infinitely many delays

Summary In this paper we consider the initial-value problems: (P ₁ )X(t)=(AX)(t) for t>0, X(0+)=I, X(t)=0 for t<0 and (P ₂ ) Y(t)=(QY)(t) for t>0, Y(0+)=I, Y(t)=0 for t<0, where A and Q are linear specified operators, I and0 — the identity and null matrices of order n, and X(t), Y(t) are unknown functions whose values are square matrices of order n. Sufficient conditions are established under which the problems (P ₁ ) and (P ₂ ) have the same unique solution, locally summable on the half-axis t ⩾0. Using this fact and some properties of the Laplace transform we find a new proof for the variation of constants formula given in[1, 2]. On the basis of this formula we derive certain results concerning a class of integrodifferential systems with infinite delay. Entrata in Redazione il 2 marzo 1977.     

Vector bundles with infinitely many souls

We construct the first examples of manifolds, the simplest one being , which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.

     

Markets with countably infinitely many traders

In this paper we will show how to construct, in a canonical way, a continuous economy in the sense of R. J. Aumann [1]. This construction is based on W. Hildenbrand's [3] definition of “pure competition” for an exchange economy with countably infinitely many participants.