Maschke's averaging lemma for abelian groups

Statement
Suppose $$G$$ is a finite group and $$V$$ is an abelian group such that the order of $$G$$ is invertible in $$V$$ (in other words, the map $$v \mapsto |G|v$$ is bijective).

Suppose we have an action of $$G$$ on $$V$$ by automorphisms. Then, if $$W$$ is a direct factor of $$V$$ that is invariant under the $$G$$-action, there exists a complement $$W'$$ to $$W$$ in $$V$$ that is also invariant under the $$G$$-action.

In the particular case where $$V$$ is an elementary abelian group, we get the usual Maschke's lemma for prime fields.

Related facts

 * Maschke's averaging lemma: Here, the Abelian group is the additive group of a field whose characteristic does not divide the order of the group. The condition on the characteristic of the field.

Proof
Given:

Textbook references

 * , Page 69, Theorem 3.2