Abelian p-group with indecomposable coprime automorphism group is homocyclic

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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.


Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, More info, Page 176, Theorem 2.2, Section 5.2 (p'-automorphisms of Abelian p-groups)