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

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

Groups with many subgroups which are commensurable with some normal subgroup

A subgroup H of a group G is called commensurable with a normal subgroup (cn) if there is N C G such that |HN/(H ∩ N)| is finite. We characterize generalized radical groups G which have one of the following finiteness conditions: (A) the minimal condition on non-cn subgroups of G; (B) the non-cn subgroups of G fall into finitely many conjugacy classes; (C) the non-cn subgroups of G have finite ran

Algebraic entropy in locally linearly compact vector spaces

We introduce algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as a natural extension of the algebraic entropy for endomorphisms of discrete vector spaces studied in Giordano Bruno and Salce (Arab J Math 1:69\u201387, 2012). We show that the main properties continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem

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 $|X/(X\cap \varphi(X))|<\infty$ for each $X\le A$. We show that this implies the above one, provided $A$ has finite torsion-free rank

On groups whose subnormal subgroups are inert

A subgroup $H$ of a group $G$ is said to be inert if $H\cap H^g$ has finite index in both $H$ and $H^g$ for any $g\in G$. We study hyper-(abelian or finite) groups in which subnormal subgroups are inertial

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

A group of generalized finitary automorphisms of an abelian group

We study the group IAut(A) generated by the inertial automorphisms of an abelian group A, that is, automorphisms g with the property that each subgroup H of A has finite index in the subgroup generated by H and H^g. Clearly, IAut(A) contains the group FAut(A) of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that IAut(A) is (locally finite)-by-abelian. In this paper, we show that IAut(A) is also metabelian-by-(locally finite). More precisely, IAut.A has a normal subgroup G such that IAut(A)/G= is locally finite and the derived subgroup G' is an abelian periodic subgroup all of whose subgroups are normal in G. In the case when A is periodic, IAut(A) turns out to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of IAut(A)

Inertial automorphisms of an abelian group

An automorphisms $\gamma$ of a group is inertial if X\cap X^\g has finite index in both $X$ and X^\g for any subgroup $X$. We study inertial automorphisms of abelian groups and give characterization of them. In particular, if the group is periodic they have property that\ud $X^{}/X_{}$ is bounded. We also study finitely generated groups of inertial automorphisms