Characterization of the automorphisms of an abelian torsion group having the weakly extension property

Abstract

In this paper, we show that if an automorphism α of an abelian torsion group, which is in fact a direct sum of its p-components, has the weak extension property then α = πid Ap + ρ, where p is a prime number, π is an invertible p-adic number and ρ ∈ Hom(A p , A 1 p ) with A 1 p is the first Ulm subgroup of the p-component A p of A.

Coonstruction of an automorphism of an abelian group that satisfies the property of the weak extension without satisfying the property of the extension

“…By theorem 1.1 [10], the automorphism ϕ satisfy the property of the weak extension. By against the automorphism ϕ does not satisfy the property of the extension because the only automorphisms satisfying the property of extension in the category of abelian groups reduced are: ±id, see [12].…”

“…It is not yet known whether this result is true for algebras (of finite dimensions) over a field. The automorphisms of abelian groups having the extension property in the category of abelian groups are characterized in [12] and the automorphisms of abelian groups having the weakly extension property in the category of abelian groups are characterized in [10].…”

A note on Wedderburn and Hasse theorems

“…A well-known proof was given by Ted Kaczynski in 1964 [9] and where the author acknowledged the earlier historical proofs. Since we are more interested in our team to rely on Abelian group-theoretic proofs for many problems as in [23,5,1,24,2], we also suggest to read the non-detailed but interesting proof done by Hans Zassenhaus in 1953 [22]. Following this same way of reasoning while combining some elements of the proof approaches in [6] as well as the one in [17], we succeed here to show how is it possible to introduce interchangeably the index of the subgroup or the order of the orbit by either defining a stabilizer or centralizer.…”