Fully inert subgroups of divisible Abelian groups

Dikranjan, Dikran; Bruno, Anna Giordano; Salce, Luigi; Virili, Simone

doi:10.1515/jgt-2013-0014

Cited by 29 publications

(26 citation statements)

References 9 publications

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“…In [5] [4] and [17] for inner automorphisms, one can introduce a smaller class of subgroups of an Abelian group G, letting φ range in the ring End(G) of all endomorphisms of G. Namely, call a subgroup H of an Abelian group G fully inert if H is φ-inert for every endomorphism φ of G. The family φ∈End(G) I φ (G) of all fully inert subgroups of G contains all finite subgroups, all finite-index subgroups and all fully invariant subgroups of G. Fully inert subgroups are investigated in the papers [8] and [10], with particular attention paid to the Abelian groups which are fully inert in divisible groups and free groups respectively.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic algebraic entropy

Dikranjan

Bruno

Salce

et al. 2015

Journal of Pure and Applied Algebra

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The new notion of intrinsic algebraic entropy (ent) over tilde of endomorphisms of Abelian groups is introduced and investigated. The intrinsic algebraic entropy is compared with the algebraic entropy, a well-known numerical invariant introduced in the sixties and recently deeply studied also in its relations to other fields of Mathematics. In particular, it is shown that the intrinsic algebraic entropy and the algebraic entropy coincide on endomorphisms of torsion Abelian groups, and their precise relation is clarified in the torsion-free case. The Addition Theorem and the Uniqueness Theorem are also proved for ent, in analogy with similar results for the algebraic entropy. Furthermore, a relevant connection of eat to the algebraic entropy of a continuous endomorphism of a locally compact Abelian group G is pointed out; this allows for the computation of the algebraic entropy in case G is totally disconnected

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Section: Introductionmentioning

confidence: 99%

Intrinsic algebraic entropy

Dikranjan

Bruno

Salce

et al. 2015

Journal of Pure and Applied Algebra

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“…The generalization of the inert subgroup via homomorphisms has also attracted the attention of other authors. In [6], Dikranjan et al gave such a generalization by endomorphisms. Namely, they defined the fully inert subgroups.…”

Section: Theorem 14 a Finitely Generated Group G Is Strongly Autinermentioning

confidence: 99%

“…However, a strongly autinertial group need not satisfy P # . For example, (Q, +) is strongly autinertial (see [6]). On the other hand, it is a characteristically simple group and it has no proper finite index subgroup.…”

Section: Theorem 14 a Finitely Generated Group G Is Strongly Autinermentioning

confidence: 99%

On strongly autinertial groups

2018

Turk J Math

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A subgroup X of G is said to be inert under automorphisms (autinert) if |X : X α ∩ X| is finite for all α ∈ Aut(G) and it is called strongly autinert if | < X, X α >: X| is finite for all α ∈ Aut(G). A group is called strongly autinertial if all subgroups are strongly autinert. In this article, the strongly autinertial groups are studied. We characterize such groups for a finitely generated case. Namely, we prove that a finitely generated group G is strongly autinertial if and only if one of the following hold:where F is a finite subgroup of G and ⟨a⟩ is a torsion-free subgroup of G.Moreover, in the preliminary part, we give basic results on strongly autinert subgroups.

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“…As in [6] and [7], an endomorphism ϕ of an abelian group A (from now on always in additive notation) is said (right-) inertial iff:…”

Section: Introductionmentioning

confidence: 99%

Inertial endomorphisms of an abelian group

Dardano

Rinauro

2014

Annali di Matematica

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We describe inertial endomorphisms of an abelian group A, that is endomorphisms\ud ϕ with the property |(ϕ(X) + X)/X| < ∞ for each X ≤ A. They form a ring\ud I E(A) containing the ideal F(A) formed by the so-called finitary endomorphisms, the ring\ud of power endomorphisms and also other non-trivial instances.We show that the quotient ring\ud I E(A)/F(A) is commutative. Moreover, inertial invertible endomorphisms form a group,\ud provided A has finite torsion-free rank. In any case, the group I Aut(A) they generate is\ud commutative modulo the group FAut (A) of finitary automorphisms, which is known to be\ud locally finite. We deduce that I Aut(A) is locally-(center-by-finite). Also, we consider the\ud lattice dual property, that is |X/(X ∩ϕ(X))| < ∞for each X ≤ A and show that this implies\ud the above one, provided A has finite torsion-free rank

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Fully inert subgroups of divisible Abelian groups

Cited by 29 publications

References 9 publications

Intrinsic algebraic entropy

Intrinsic algebraic entropy

On strongly autinertial groups

Inertial endomorphisms of an abelian group

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