Algebraic Entropies for Abelian Groups with Applications to the Structure of Their Endomorphism Rings: A Survey

Goldsmith, Brendan; Salce, Luigi

doi:10.1007/978-3-319-51718-6_7

Cited by 12 publications

(8 citation statements)

References 49 publications

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“…Here we recall the following counterpart of the Algebraic Youzvinski formula obtained in [52]. Further details and results can be found in [52] and in the survey paper [74].…”

Section: Applications To Algebraic Entropy Of Group Endomorphismsmentioning

confidence: 99%

“…The concept of inert subgroup has many applications in different areas of algebra. In this paper we try to collect some of them and continue the work done by previous surveys as [5,71,74] even if we are conscious that we cover only a part of the subject. Many of these results have been presented at several meetings "Ischia Group Theory", an environment that stimulated many interesting and fruitful discussions.…”

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confidence: 99%

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Inertial Properties in Groups

Dardano¹,

Dikranjan²,

Rinauro³

2017

Preprint

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Let G be a group and ϕ be an endomorphism of G. A subgroup H of G is called ϕ-inert if H ϕ ∩ H has finite index in the image H ϕ . The subgroups that are ϕ-inert for all inner automorphisms of G are widely known and studied in the literature, under the name inert subgroups.The related notion of inertial endomorphism, namely an endomorphism ϕ such that all subgroups of G are ϕ-inert, was introduced in [34] and thoroughly studied in [37,35]. The "dual" notion of fully inert subgroup, namely a subgroup that is ϕ-inert for all endomorphisms of an abelian group A, was introduced in [51] and further studied in [25,54,76].The goal of this paper is to give an overview of up-to-date known results, as well as some new ones, and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra. We survey on classical and recent results on groups whose inner automorphisms are inertial. Moreover, we show how inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces, and can be helpful for the computation of the algebraic entropy of continuous endomorphisms. PreliminariesClearly normal and finite subgroups are inert. Many generalizations of the notion of normality have appeared in the literature (see for example [71]). The concept of inert subgroup allows to put many of them in a common framework. It seems to have been introduced in 1993 in papers of Belyaev [3,4] as a tool in the investigation of infinite simple groups. In a survey paper on this subject [5] Belyaev gives credit to Kegel for coining the term inert subgroup. The concept of inert subgroup tacitly involves inner automorphisms of a group and has been therefore extended to general automorphisms ([34]) in 2012. Then in [51] the concept has been extended to general endomorphisms of abelian groups. Let us give a basic definition which reveals to be a powerful tool. One may replace the relation of invariance of a subgroup with respect to an endomorphism by a weaker condition which is trivially satisfied by all finite subgroups. Definition 1.1. Let ϕ be an endomorphism and H a subgroup of a group G. Then H is called ϕ-inert if H ϕ ∩ H has finite index in the image H ϕ .

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“…Here we recall the following counterpart of the Algebraic Youzvinski formula obtained in [52]. Further details and results can be found in [52] and in the survey paper [74].…”

Section: Applications To Algebraic Entropy Of Group Endomorphismsmentioning

confidence: 99%

mentioning

confidence: 99%

Inertial Properties in Groups

Dardano¹,

Dikranjan²,

Rinauro³

2017

Preprint

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“…In the next remark and in the following we deal with the algebraic entropy of endomorphisms of p-groups. For an illustration of the notion of algebraic entropy and its properties we refer to [8] and to our survey paper [13]. We just recall here that the algebraic entropy of an endomorphism φ of an abelian group G, denoted by ent(φ), is an invariant (actually, a length function on the category of Z[X ]-modules), which measures by means of nonnegative real numbers and the symbol ∞ the behaviour of the discrete dynamical system obtained by the powers of φ.…”

Section: Theorem 24 Let G Be a Separable Semi-standard P-group Such That In The Pierce Decomposition End(g) = A ⊕ E S (G) A Is The Complementioning

confidence: 99%

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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“…The class N of narrow groups was introduced in [15] as an example of a large class of abelian groups G such that every endomorphism of G has adjoint algebraic entropy zero, and was also highlighted in [14]. The adjoint algebraic entropy is the natural dual of the algebraic entropy for endomorphisms of abelian groups, deeply investigated in [13,18] (see also the survey [23]). In [15] the following characterizations of narrow groups are presented, involving the family C(G) of all subgroups of nite index of G. as a continuous image of the compact group K under the multiplication by n).…”

Section: Orsatti Groups Groups With Nite Ranks and Narrow Groupsmentioning

confidence: 99%

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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Algebraic Entropies for Abelian Groups with Applications to the Structure of Their Endomorphism Rings: A Survey

Cited by 12 publications

References 49 publications

Inertial Properties in Groups

Inertial Properties in Groups

Abelian p-groups with minimal full inertia

On uniformly fully inert subgroups of abelian groups

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