Sequences of Endomorphism Groups of Abelian Groups

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite $p$-ranks. For an Abelian group $A$, a sequence of groups $(A_n)$ is considered, where $A_0=A$ and $A_{n+1}=\operatorname{End}A_n$. It is shown that, if all $p$-ranks of the group $A$ are finite, then this sequence can stabilize either after $A_0$ or after $A_1$.