Splitting the Automorphism Group of an Abelian P-group

Let G be an abelian p-group sum of finite homocyclic groups Gi. Here, we determine in which cases the automorphism group of G splits over kerσ, where σ :Aut(G) → ∏ iAut(Gi/pGi) is the natural epimorphism. 1. Preliminaries Throghout this paper, p is an arbitrary prime and r is a fixed ordinal number. Let G = ⊕i≤rGi be an abelian p-group, such that Gi is an homocyclic group of exponent pi and finite p-rank ri, with ni < ni+1 for all i. It is known that E the endomorphism ring of G is isomorphic to the ring E(G) of all row finite r × r-matrices (Aij) where Aij ∈ Hom(Gi, Gj). We denote by A(H), the automorphism group of a group H and consider A(H) as the group of units of the endomorphism ring E(H). Let σ be the natural epimorphism of A(G) onto the product of the A(Gi/pGi). We have the following exact sequence (see [2], page 256) (1) 1 → kerσ → A(G) → ∏