orbit-stabilizer theorem
Suppose that is a group acting (http://planetmath.org/GroupAction) on a set .
For each , let be the orbit of ,
let be the stabilizer of ,
and let be the set of left cosets
of .
Then for each the function
defined by is a bijection.
In particular,
and
for all .
Proof:
If is such that for some ,
then we have , and so ,
and therefore .
This shows that is well-defined.
It is clear that is surjective.
If , then for some ,
and so .
Thus is also injective
.
Title | orbit-stabilizer theorem |
---|---|
Canonical name | OrbitstabilizerTheorem |
Date of creation | 2013-03-22 12:23:10 |
Last modified on | 2013-03-22 12:23:10 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 22 |
Author | yark (2760) |
Entry type | Theorem![]() |
Classification | msc 20M30 |