Inertial Automorphisms of an Abelian Group

Abstract

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). In particular, IAut(A) has a normal subgroup Γ such that IAut(A)/Γ is locally finite and Γ ′ is an abelian periodic subgroup whose all subgroups are normal in Γ. In the case when A is periodic, IAut(A) results 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) in some other cases for A.If A is any periodic abelian group, then PAut(A) is the cartesian product of all theAccording to [10], an automorphism γ is called an (invertible) multiplication of A if and only if it is a power automorphism of A, if A is periodic, or -when A is non-periodicthere exist coprime integers m, n such that (na)γ = ma, for each a ∈ A. In the latter case, we have mnA = A and A π(mn) = 0 and -with an abuse of notation-we will write γ = m/n. We warn that we are using the word "multiplication" in a way different from [14]. Invertible multiplications of A form a subgroup which is a central subgroup of Aut(A).

Intrinsic algebraic entropy

“…This suitable notion of subgroup (see Definition 1.1) is inspired by that of inert subgroup introduced by Belyaev in [3] for inner automorphisms of non-commutative groups and investigated in that setting in [4,17,5]. (Recall that a subgroup H of G is inert if [H : H ∩ φ(H)] is finite for every inner automorphism φ of G; Belyaev [3] gives credit to Kegel for coining the term "inert subgroup".…”

Inertial endomorphisms of an abelian group

“…We treat now inertial and LIN endomorphisms of a non-periodic abelian group. Next, proposition generalizes Theorem 3 of [3] to endomorphisms. Even if the statement has interest in itself, we will regard it as a lemma for Theorem A.…”

“…On the other hand, from results in [3], we have that for an automorphism ϕ of a periodic abelian group A properties LIN and RIN (i.e., inertial) are equivalent, that is each subgroup is commensurable with its image. Therefore, in [3] by inertial we meant LIN and RIN, while here by inertial we mean just RIN.…”

“…Therefore, in [3] by inertial we meant LIN and RIN, while here by inertial we mean just RIN. Next, corollary to Theorem A below investigates how these properties are related.…”

“…Clearly, in both cases, Γ is formed by automorphisms which are inertial. Then, in [3], we put both pictures in the same framework by characterizing finitely generated groups Γ of automorphisms which are inertial and have inertial inverse. In Propositions 2.2 and 2.3 of this paper, we give a characterization also in the more general setting of endomorphisms.…”