PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (212)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (38)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
orbit-stabilizer theorem (Theorem)

Suppose that $ G$ is a group acting on a set $ X$. For each $ x\in X$, let $ Gx$ be the orbit of $ x$, let $ G_x$ be the stabilizer of $ x$, and let $ {\cal L}_x$ be the set of left cosets of $ G_x$. Then for each $ x\in X$ the function $ f\colon Gx\to{\cal L}_x$ defined by $ gx\mapsto gG_x$ is a bijection. In particular,

$\displaystyle \vert Gx\vert = [G:G_x] $
and
$\displaystyle \vert Gx\vert\cdot\vert G_x\vert = \vert G\vert $
for all $ x\in X$.

Proof:
If $ y\in Gx$ is such that $ y=g_1x=g_2x$ for some $ g_1,g_2\in G$, then we have $ g_2^{-1}g_1x=g_2^{-1}g_2x=1x=x$, and so $ g_2^{-1}g_1\in G_x$, and therefore $ g_1G_x=g_2G_x$. This shows that $ f$ is well-defined.

It is clear that $ f$ is surjective. If $ gG_x = g'G_x$, then $ g = g'h$ for some $ h \in G_x$, and so $ gx = (g'h)x= g'(hx) = g'x$. Thus $ f$ is also injective.



"orbit-stabilizer theorem" is owned by yark. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

Log in to rate this entry.
(view current ratings)

Cross-references: injective, surjective, clear, well-defined, bijection, function, left cosets, stabilizer, orbit, group
There are 8 references to this entry.

This is version 19 of orbit-stabilizer theorem, born on 2002-02-19, modified 2008-05-09.
Object id is 2173, canonical name is OrbitStabilizerTheorem.
Accessed 8477 times total.

Classification:
AMS MSC: 20M30 (Group theory and generalizations :: Semigroups :: Representation of semigroups; actions of semigroups on sets)
Pending Errata and Addenda
None.
[ View all 8 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)