Solitary subgroups of Abelian groups

Since solitary subgroups of (infinite) Abelian groups are precisely the strictly invariant subgroups which are co-Hopfian (as groups), and strictly invariant subgroups turn out to be strongly invariant for large classes of Abelian groups we determine the solitary subgroups for these classes of groups.     

Fully transitive torsion-free Abelian groups

Translated from Algebra i Logika, Vol. 29, No. 5, pp. 549–560, September–October, 1990.     

Tight subgroups in torsion-free Abelian groups

LetX be a torsion-free abelian group. We study the class of all completely decomposable subgroups ofX which are maximal with respect to inclusion. These groups are called tight subgroups ofX and we state sufficient conditions on a subgroup to be tight. In particular we consider tight subgroups of bounded completely decomposable groups. For those we show that every regulating subgroup is tight and we characterize the tight subgroups of finite index in almost completely decomposable groups. The second author was supported by a MINERVA fellowship.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

On strongly invariant subgroups of Abelian groups

Mathematical Notes - It is shown that every homogeneous separable torsion-free group is strongly invariant simple (i.e., has no nontrivial strongly invariant subgroups) and, for a completely...     

On projective invariant subgroups of Abelian groups

The properties of projective invariant subgroups are studied. The structure of these subgroups in nonreduced groups is described. The conditions under which projective invariant subgroups are fully invariant are considered.     

Torsion-free Abelian groups with cyclic p-basic subgroups

We investigate the structure of indecomposable torsion-free Abelian groups all of whose p-basic subgroups are cyclic, and also the structure of the groups and rings of endomorphisms of such groups. We prove the existence of a torsion-free Abelian group of countable rank with cyclic p-basic subgroups which has no indecomposable nonzero direct summands.Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 805–813, December, 1976.     

Abelian Carter subgroups in finite permutation groups

We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.     

On finite groups with cyclic Abelian subgroups

Solvable and minimal unsolvable finite groups with cyclic Abelian subgroups are constructively described. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 739–741, May, 1998.     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.     

On locally free Abelian groups

The following result is proved in the paper. An Abelian group A is Lw₁, w-equivalent to the free Abelian group of countable rank if and only if it is a countably free Abelian group.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 121–126.Original Russian Text Copyright © 2005 by E. G. Sklyarenko.This revised version was published online in April 2005 with a corrected issue number.     

On fully inert subgroups of completely decomposable groups

Mathematical Notes - The completely decomposable torsion-free Abelian groups with finitely many homogeneous components for which every fully inert subgroup is commensurable with a fully invariant...     

Maximal orders of Abelian subgroups in finite simple groups

We bring out upper bounds for the orders of Abelian subgroups in finite simple groups. (For alternating and classical groups, these estimates are, or are nearly, exact.) Precisely, the following result, Theorem A, is proved. Let G be a non-Abelian finite simple group and G L₂ (q), where q=p^t for some prime number p. Suppose A is an Abelian subgroup of G. Then |A|³<|G|. Our proof is based on a classification of finite simple groups. As a consequence we obtain Theorem B, which states that a non-Abelian finite simple group G can be represented as ABA, where A and B are its Abelian subgroups, iff G≌ L₂(2^t) for some t ≥ 2; moreover, |A|-2^t+1, |B|=2^t, and A is cyclic and B an elementary 2-group. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 131–160, March–April, 1999.     

Countable mixed Abelian groups with very nice full-rank subgroups

Measure concentration arguments are applied to get a power-type estimate for the dimension of almostl _p subspaces of isomorphs ofl _p ⁿ and for the length of almost-symmetric sequences under a nonlinear-type condition. Supported by the Fund for Basic Research Administered by the Israel Academy of Sciences and Humanities.