# Abelian p-group with indecomposable coprime automorphism group is homocyclic

Suppose $P$ is an Abelian group of prime power order and $A \le \operatorname{Aut}(P)$ is a group of order relatively prime to $p$, such that $A$ acts indecomposably on $P$. In other words, $P$ cannot be expressed as an internal direct product of $A$-invariant subgroups. Then, $P$ is a homocyclic group.
• Finite Groups by Daniel Gorenstein, ISBN 0821843427, More info, Page 176, Theorem 2.2, Section 5.2 ($p'$-automorphisms of Abelian $p$-groups)