Maschke's averaging lemma for abelian groups: Difference between revisions
(New page: ==Statement== 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 <math>p</math>-group, and we ha...) |
No edit summary |
||
| Line 1: | Line 1: | ||
==Statement== | ==Statement== | ||
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 | ==Statement for Abelian groups=== | ||
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. | In the particular case where <math>V</math> is elementary Abelian, we get the usual [[Maschke's lemma]] for prime fields. | ||
==Related facts== | |||
* [[Maschke's averaging lemma]]: Here, | |||
==References== | ==References== | ||
===Textbook references=== | ===Textbook references=== | ||
* {{booklink-proved|Gorenstein}}, Page 69, Theorem 3.2 | * {{booklink-proved|Gorenstein}}, Page 69, Theorem 3.2 | ||
Revision as of 22:26, 17 October 2008
Statement
Statement for Abelian groups=
Suppose is a finite group, whose order is relatively prime to a prime . Suppose is an Abelian group, and 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 elementary Abelian, we get the usual Maschke's lemma for prime fields.
Related facts
- Maschke's averaging lemma: Here,
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, More info, Page 69, Theorem 3.2