Maschke's averaging lemma for abelian groups: Difference between revisions

Latest revision as of 19:31, 9 July 2011

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

@@ Line 1: / Line 1: @@
 ==Statement==
-==Statement for Abelian groups===
+Suppose <math>G</math> is a [[finite group]] and <math>V</math> is an [[abelian group]] such that the order of <math>G</math> is invertible in <math>V</math> (in other words, the map <math>v \mapsto |G|v</math> is bijective).
-Suppose <math>G</math> is a [[finite group]], whose order is relatively prime to a prime <math>p</math>. Suppose <math>V</math> is an Abelian group, and we have an action of <math>G</math> on <math>V</math> by automorphisms. Then, if <math>W</math> is a [[direct factor]] of <math>V</math> that is invariant under the <math>G</math>-action, there exists a complement <math>W'</math> to <math>W</math> in <math>V</math> that is also invariant under the <math>G</math>-action.
-In the particular case where <math>V</math> is elementary Abelian, we get the usual [[Maschke's lemma]] for prime fields.
+Suppose we have an action of <math>G</math> on <math>V</math> by automorphisms. Then, if <math>W</math> is a [[direct factor]] of <math>V</math> that is invariant under the <math>G</math>-action, there exists a complement <math>W'</math> to <math>W</math> in <math>V</math> that is also invariant under the <math>G</math>-action.
+In the particular case where <math>V</math> is an [[elementary abelian group]], we get the usual [[Maschke's lemma]] for prime fields.
 ==Related facts==
-* [[Maschke's averaging lemma]]: Here,
+* [[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===
 * {{booklink-proved|Gorenstein}}, Page 69, Theorem 3.2