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:

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 69, Theorem 3.2