Maschke's averaging lemma for abelian groups
From Groupprops
Statement
Suppose is a finite group and is an abelian group such that the order of is invertible in (in other words, the map is bijective).
Suppose we have an action of on by automorphisms. Then, if is a direct factor of that is invariant under the -action, there exists a complement to in that is also invariant under the -action.
In the particular case where 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