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 |