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It is shown that a generalised radical group has no chain of non-pronormal subgroups with the same order type as the set R of the real numbers if and only if either the group is minimax or all subgroups are pronormal. [ABSTRACT FROM AUTHOR]

We study soluble groups G in which each subnormal subgroup H with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup N such that H ∩ N has finite index in both H and N. We show that if such a G is periodic, then all subnormal subgroups are commensurable with a normal subgroup, provided either the Hirsch-Plotkin radical of G has infinite rank or G is nilpotent-by-abelian (and has infinite rank). [ABSTRACT FROM AUTHOR]

We study groups in which each subnormal subgroup is commensurable with a normal subgroup. Recall that two subgroups H and K are termed commensurable if H ∩ K has finite index in both H and K. Among other results, we show that if a (sub)soluble group G has the above property, then 𝐺 is finite-by-metabelian, i.e., G ′′ is finite. [ABSTRACT FROM AUTHOR]

We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups of G/N are abelian-by-finite. Then we apply this result to the study of groups which are minimal-non- P also for related group properties P . Finally we see how for locally (soluble-by-finite) groups of infinite rank, it is enough to restrict attention to the proper subgroups with infinite rank. [ABSTRACT FROM AUTHOR]

Let G be a group and pp be an endomorphism of G. A subgroup H of G is called φ-inert if Hφ ∩ H has finite index in the image Hφ. 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. [ABSTRACT FROM AUTHOR]

We study the group IAut(A) generated by the inertial automorphisms of an abelian group A, that is, automorphisms γ with the property that each subgroup H of A has finite index in the subgroup generated by H and Hγ. 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 Γ such that IAut(A)/Γ is locally finite and the derived subgroup Γ' is an abelian periodic subgroup all of whose subgroups are normal in Γ. 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). [ABSTRACT FROM AUTHOR]

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 $$IE(A)$$ containing the ideal $$F(A)$$ formed by the so-called finitary endomorphisms, the ring of power endomorphisms and also other non-trivial instances. We show that the quotient ring $$IE(A)/F(A)$$ is commutative. Moreover, 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 $$X\le A$$ and show that this implies the above one, provided $$A$$ has finite torsion-free rank. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called inert if, for each g ∈ G, the index of H ∩ Hg in H is finite. We give a classification 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. [ABSTRACT FROM AUTHOR]

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. We obtain a satisfactory description modulo the ideal of finitary endomorphisms. Also the corresponding problem for vector spaces is considered. [ABSTRACT FROM AUTHOR]

We address the question of the subnormality of a subgroup H in a group G when H is subnormal in two subgroups U and V such that G = (U, V). In particular, we consider the case in which G is nilpotent-by-abelian and the case in which G = UV is factorized by U and V. [ABSTRACT FROM AUTHOR]

We consider groups Γ of automorphisms of a group G acting by means of power automorphisms on the factors of a normal series in G with length m. We show that [ G, Γ] is nilpotent with class at most m and that this bound is best possible. Moreover, such a Γ is parasoluble with paraheight at most 1/2 m(m+3)+1, provided Γ′ is periodic. We give best possible bound in the case where the series is a central one. [ABSTRACT FROM AUTHOR]