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 soluble groups whose subnormal subgroups are inert

A subgroup H of a group G is called inert if, for each g in G, the index of H intersection H^g in H is finite. We give a classication of soluble-by-finite groups G in which subnormal subgroups are inert in the cases where G has no nontrivial torsion normal subgroups or G is finitely generated

A class of groups with inert subgroups

Two subgroups H and K of a group are commensurable iff their intersection has finite index in both H and K. We prove that hyper-(abelian or finite) groups with finite abelian total rank in which every subgroup is commensurable to a normal one are finite-by-abelian-by-finite

On the ring of inertial endomorphisms of an abelian p-group

An endomorphisms $\varphi$ of a group $G$ is said inertial if $\forall H\le G$ \ \ $|\varphi(H):(H\cap \varphi(H))|<\infty$. Here 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

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

Some lattice properties of virtually normal subgroups

The set $vn( G)$ of subgroups with only finitely many conjugates in a group G is a sublattice of the lattice of all subgroups of G. Here groups G are studied for which $vn( G)$ is decomposable, complemented and relatively complemented

Modularity and distributivity in the lattice of virtually normal subgroups

We give a characterization of groups with modular or distributive lattice of virtually normal subgroups, that is subgroups with only finitely many conjugates. We also show that in many cases the distributivity of this lattice implies that all virtually normal subgroups are normal