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'orbit-stabilizer theorem'
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| Title of object: |
orbit-stabilizer theorem |
| Canonical Name: |
OrbitStabilizerTheorem |
| Type: |
Theorem |
| Created on: |
2002-02-19 06:41:46-05 |
| Modified on: |
2003-02-11 17:19:31.796183-05 |
Preamble:
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%\usepackage{psfrag}
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Content:
Given a group action $G$ on a set $X$, define $Gx$ to be the orbit of $x$ and $G_x$ to be the set of stabilizers of $x$. For each $x \in X$ the correspondence $\operatorname{g}(x) \rightarrow gG_x$ is a bijection between $Gx$, and the set of left cosets of $G_x$
A famous corollary is that
$|\operatorname{G}(x)|.|G_x| = |G| \quad \forall x\in X$ |
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