Exercise 5, § 4

Let M be a module of finite length, and let $M =\oplus_{i\in I}M_i$ be a decomposition of M into a direct sum of indecomposable submodules. Let $J\subset I$ be an equivalence class for the relation “the modules $M_i$ and $M_j$ are isomorphic” between $i$ and $j$; we set $M_J=\oplus_{i\in J}M_i$. Give an example to show that $M_J$ is not necessarily stable under every automorphism of M (use Exercise 4 with M taken to be a $\mathbf{Z}$-module of finite length).