On uniformly fully inert subgroups of abelian groups

Abstract

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.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.

Abelian p-groups with minimal full inertia

“…In Sect. 5 we prove that the new examples, furnished in the preceding sections, of fully inert subgroups which fail to be commensurable with fully invariant subgroups are not uniformly fully inert, thus giving further evidence of the likely truth of Conjecture 1.6 in [6], which states that every uniformly fully inert subgroup of an arbitrary group is commensurable with a fully invariant subgroup.…”

“…Thus, from now on, when dealing with fully inert subgroups H of a direct sum of cyclic groups i∈I G i , we will assume that H = i∈I H i , with H i ≤ G i for all i. This situation is expressed by saying that H is a box-like subgroup of G in [7], where this terminology was introduced and this notion was used for direct sums of divisible groups; it was used also more recently in [3], [6] and [14].…”

“…In [6,Proposition 6.2] it was proved that the socle B[ p] is not uniformly fully inert, that is, the cardinalities of the quotients…”

“…This fact, together with many other exhibited examples for different families of abelian groups, tempted the authors of [6] to formulate the conjecture (already presented in [5]) that every uniformly fully inert subgroup of a group G is commensurable with a fully invariant subgroup. Following [6], we denote by I u (G) the subset of I(G) consisting of the uniformly fully inert subgroups, that is, of the subgroups H such that |(φ H + H )/H | ≤ N for all endomorphisms φ and for a fixed positive integer N . In [6] it was proved that Inv˜(G) ⊆ I u (G).…”

“…A subgroup commensurable with a fully inert subgroup is also fully inert, so, in particular, a subgroup H commensurable with a fully invariant subgroup is fully inert. Following [5] and [6], denote by Inv(G) the set of fully invariant subgroups of G and by Inv˜(G) the set of subgroups of G which are commensurable with fully invariant subgroups.…”