Maschke's averaging lemma for abelian groups

Statement

Suppose $G$ is a finite group, whose order is relatively prime to a prime $p$. Suppose $V$ is an Abelian $p$-group, and 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 $W^{\prime }$ to $W$ in $V$, that is also invariant under the $G$-action.

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

References

Textbook references

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