Fully inert subgroups of Abelian p-groups

Goldsmith, Brendan; Salce, Luigi; Zanardo, Paolo

doi:10.1016/j.jalgebra.2014.07.021

Cited by 24 publications

(28 citation statements)

References 5 publications

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“…Thus, a fortiori, the equality Inv˜(G) = Iu(G) holds as well. These facts are the main results in the three papers [19], [24] and [25], respectively.…”

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

confidence: 61%

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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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“…Thus, a fortiori, the equality Inv˜(G) = Iu(G) holds as well. These facts are the main results in the three papers [19], [24] and [25], respectively.…”

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

confidence: 61%

“…On the other hand, an example of a separable p-group G which admits a subgroup H ∈ I(G) \ Inv˜(G) is provided in [24]. Furthermore, an example of a torsion-free Jp-module A admitting a submodule K ∈ I(A) \ Inv˜(A) is provided in [25].…”

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

confidence: 99%

On uniformly fully inert subgroups of abelian groups

Dardano

Dikranjan

Salce

2020

Topological Algebra and Its Applications

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“…This entropy is a generalization of the algebraic entropy ent (as it coincides with ent on torsion abelian groups) but does not vanish on torsion-free abelian groups. It involves the concept of inert subgroup, that was deeply investigated in several papers (see [27,28,43,49,70]).…”

Section: Final Remarksmentioning

confidence: 99%

Entropy on normed semigroups (towards a unifying approach to entropy)

Dikranjan

Bruno

2019

Dissertationes Math.

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We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, various forms of topological entropy, several notions of algebraic entropy, and two forms set-theoretic entropy. We take into account only discrete dynamical systems, that is, pairs (X, T ), where X is the underlying space (e.g., a probability space, a compact topological space, a group, a set) and T : X → X is a transformation of X (e.g., a measure preserving transformation, a continuous selfmap, a group homomorphism, a selfmap). We see entropies as functions h : X → R + , associating to each flow (X, φ) of a category X either a non negative real number or ∞.First, we introduce the notion of semigroup entropy h S : S → R + , which is a numerical invariant attached to endomorphisms of the category S of normed semigroups. Then, for a functor F : X → S from any specific category X to S, we define the functorial entropy h F :Clearly, this entropy h F inherits many of the properties of h S , depending also on the properties of the functor F . Motivated by this aspect, we study in detail the properties of h S . Such general scheme, using elementary category theory, permits to obtain many relevant known entropies as functorial entropies h F , for appropriately chosen categories X and functors F : X → S. For example, most of the above mentioned measure entropy, topological entropies, algebraic entropies and set-theoretic entropies are functorial entropies. Furthermore, we exploit our scheme to elaborate a common approach to establish the properties shared by those entropies that we find as functorial entropies. In this way we point out their common nature.Finally, we discuss and deeply analyze through the looking glass of our unifying approach the relations between pairs of entropies. To this end we first formalize the notion of Bridge Theorem between two entropies h 1 : X 1 → R + and h 2 : X 2 → R + with respect to a functor ε : X 1 → X 2 , taking inspiration from the theorem relating the topological and the algebraic entropy by means of Pontryagin duality. Then, for pairs of functorial entropies we use the above scheme to introduce the notion and the related scheme of Strong Bridge Theorem. It allows us to put under the same umbrella various relations between pairs of entropies (e.g., the above mentioned connection of the topological and the algebraic entropy and their relation with the set-theoretic entropy).

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“…In [7], [8], and [11], the fully inert subgroups of free groups, p-groups, respectively divisible groups were studied and described.…”

Section: Strongly Inert Subgroupsmentioning

confidence: 99%

“…Dikranian, Giordano Bruno, Goldsmith, Salce, Virili and Zanardo defined and studied fully inert subgroups of Abelian groups in [6], [7], [8], [10] and [11]. Strongly invariant subgroups of Abelian groups were defined and studied by the second named author in [4].…”

Section: Introductionmentioning

confidence: 99%