Inertial endomorphisms of an abelian group

We describe inertial endomorphisms of an abelian group $A$, that is endomorphisms $\varphi$ with the property $|(\varphi(X)+X)/X|<\infty$ for each $X\le A$. They form a ring containing multiplications, the so-called finitary endomorphisms and non-trivial instances. We show that inertial invertible endomorphisms form a group, provided $A$ has finite torsion-free rank. In any case, the group $IAut(A)$ they generate is commutative modulo the group $FAut(A)$ of finitary automorphisms, which is known to be locally finite. We deduce that $IAut(A)$ is locally-(center-by-finite). Also we consider the lattice dual property, that is that $|X/(X\cap \varphi(X))|<\infty$ for each $X\le A$. We show that this implies the above one, provided $A$ has finite torsion-free rank

On the ring of inertial endomorphisms of an abelian group

An endomorphisms $\varphi$ of an abelian group $A$ is said inertial if each subgroup $H$ of $A$ has finite index in $H+\varphi (H)$. We study the ring of inertial endomorphisms of an abelian group. Here we obtain a satisfactory description modulo the ideal of finitary endomorphisms. Also the corresponding problem for vector spaces is considered. For the characterization of inertial endomorphisms of an abelian group see arXiv:1310.4625 . The group of invertible inertial endomorphisms has been studied in arXiv:1403.4193 .Comment: see also arXiv:1310.4625 and arXiv:1403.419

Variants of theorems of Baer and Hall on finite-by-hypercentral groups

We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical theorems by R. Baer and P. Hall. Then we apply our results to groups of automorphisms of a group $G$ acting in a restricted way on an ascending normal series of $G$

On the ring of inertial endomorphismsof an abelian p-group

An endomorphisms φ of a group G is said inertial if ∀H ≤ G |φ(H) : (H ∩φ(H))| < ∞. We study the ring of inertial endomorphisms of an abelian torsion group and the group of its units. Also the case of vector spaces is considered.

Groups with the real chain condition on non-pronormal subgroups

It is shown that a gerenalised radical group has no chain of non-pronormal subgroups with the same order type as the set of the real numbers if and only if either the group is minimax or all subgroups are pronormal.Comment: arXiv admin note: text overlap with arXiv:2307.0763

On uniformly fully inert subgroups of abelian groups

AbstractIf 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

Assemblies as Semigroups

In this paper we give an algebraic characterization of assemblies in terms of bands of groups. We also consider substructures and homomorphisms of assemblies. We give many examples and counterexamples

Inertial properties in groups

‎‎Let $G$ be a group and $p$ be an endomorphism of $G$‎. ‎A subgroup $H$ of $G$ is called $p$-inert if $H^pcap H$ has finite index in the image $H^p$‎. ‎The subgroups that are $p$-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 $p$ such that all subgroups of $G$ are $p$-inert‎, ‎was introduced in cite{DR1} and thoroughly studied in cite{DR2,DR4}‎. ‎The ``dual‎" ‎notion of fully inert subgroup‎, ‎namely a subgroup that is $p$-inert for all endomorphisms of an abelian group $A$‎, ‎was introduced in cite{DGSV} and further studied in cite{Ch+‎, ‎DSZ,GSZ}‎. ‎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

ADV Perspectives in Group Theory — an open space —

ADV Perspectives in Group Theory – an open space

Groups whose non-normal subgroups are intersection of maximal subgroups

si perviene a caratterizzare, in vari termini e sotto deboli ipotesi di risolubilita', i gruppi in cui ogni sottogruppo e' intersezione di sottogruppi normali o massimali ed in particolare i gruppi risolubili nei quali i sottogruppi non-normali sono intersezione di sottogruppiu massimali