We describe inertial endomorphisms of an abelian group A, that is endomorphisms\ud ϕ with the property |(ϕ(X) + X)/X| < ∞ for each X ≤ A. They form a ring\ud I E(A) containing the ideal F(A) formed by the so-called finitary endomorphisms, the ring\ud of power endomorphisms and also other non-trivial instances.We show that the quotient ring\ud I E(A)/F(A) is commutative. Moreover, inertial invertible endomorphisms form a group,\ud provided A has finite torsion-free rank. In any case, the group I Aut(A) they generate is\ud commutative modulo the group FAut (A) of finitary automorphisms, which is known to be\ud locally finite. We deduce that I Aut(A) is locally-(center-by-finite). Also, we consider the\ud lattice dual property, that is |X/(X ∩ϕ(X))| < ∞for each X ≤ A and show that this implies\ud the above one, provided A has finite torsion-free rank

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).

An endomorphisms $\p$ of an abelian group $A$ is said inertial if each subgroup $H$ of $A$ has finite index in $H+\varphi (H)$. We\ud study the ring of inertial endomorphisms of an abelian group. We obtain a satisfactory description modulo\ud the ideal of finitary endomorphisms.\ud Also the corresponding problem for vector spaces is considered

Forna positive integer, a groupGis calledcore-nifH/H Ghas order at mostnfor every subgroupHofG(whereH Gis the normal core ofH, the largest normal subgroup ofGcontained inH). It is proved that a finite core-pp-groupGhas a normal abelian subgroup whose index inGis at mostp 2ifpâ� 2, which is the best possible bound, and at most 2 6ifp=2. © 1997 Academic Press

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