Maschke's averaging lemma for abelian groups

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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