On fully inert subgroups of completely decomposable groups

Chekhlov, A. R.

doi:10.1134/s0001434617010394

Cited by 7 publications

(8 citation statements)

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“…The following result concerning box-like subgroups was proved in [2] and [3] and is based on results in [2] and [14]. For the convenience of the reader we present a proof of part (b); this is a very slight modification of that given in [3, Lemma 2.2].…”

Section: The Canonical Projections and H Is A Fully Inert Subgroup Then H Is Commensurable With I∈i π I (H )mentioning

confidence: 95%

Abelian p-groups with minimal full inertia

Goldsmith

Salce

2021

Period Math Hung

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The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.

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Section: The Canonical Projections and H Is A Fully Inert Subgroup Then H Is Commensurable With I∈i π I (H )mentioning

confidence: 95%

Abelian p-groups with minimal full inertia

Goldsmith

Salce

2021

Period Math Hung

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“…The strict inclusion Inv˜(G) I(G) is provided also by divisible groups G whose torsion-free rank rk (G) is nite and non-zero. Moreover, Inv˜(G) I(G) is proved by a recent result by Chekhlov [6], who characterized the completely decomposable groups G of nite rank satisfying this strict inclusion (see §3.2 for more details).…”

Section: De Nition 15 a Subgroup H Of A Group G Is Uniformly Fully mentioning

confidence: 98%

“…In Section 3 we consider the chain (*) for divisible groups and completely decomposable groups recalling some known facts from [8] and [6,7], respectively.…”

Section: De Nition 15 a Subgroup H Of A Group G Is Uniformly Fully mentioning

confidence: 99%

“…The following theorem concerning the chain (*) for completely decomposable groups of nite rank combines results from three papers, namely [6][7][8] It follows from this theorem that a uniformly fully inert subgroup H of a completely decomposable groups of nite rank has nite index in its fully invariant hull H * . In Corollary 5.3 we will obtain the notably stronger result that this holds for an arbitrary torsion-free group of nite rank.…”

Section: Proposition 33 Let H Be a Non-zero Subgroup Of A Rational mentioning

confidence: 99%

“…. ⊕ An of nite rank, de ned by the property that the subgroups A i are all isomorphic; their common type is then the type t(G) of G. Recently Chekhlov extended in [7] Theorem 3.4 to completely decomposable groups of arbitrary rank, but with only a nite number of homogeneous components (i.e., the maximal homogeneous summands). First he showed that it is enough to consider reduced completely decomposable groups, and for these groups he proved the following We conjecture that one may add to the equivalent conditions of Theorem 3.6 also the equality I(G) = Iu(G), i.e., every fully inert subgroup is uniformly fully inert (see Question 7.4 at the end of the paper).…”

Section: Proposition 33 Let H Be a Non-zero Subgroup Of A Rational mentioning

confidence: 99%

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On uniformly fully inert subgroups of abelian groups

Dardano

Dikranjan

Salce

2020

Topological Algebra and Its Applications

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If H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author.A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.

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