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If 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.

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 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 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]