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Sunday, November 14, 2010

Orbit-Stabilizer Theorem

Let $G$ be a group, $X$ be a set, with $g \in G$, $x \in X$.

The number of elements in the orbit of $x$ is equal to the index of the stabilizer of $x$ in $G$:
$|\text{Orb}(x)| = [G: \text{Stab}(x)]$ ( Orbit-Stabilizer Theorem )

The total number of orbits is equal to the number of elements in $x$ fixed under an action of $g$, summed for all elements in G and finally divided by the size of $G$:
$|\text{Orb}| = \frac{1}{G} \sum_{g \in G} \text{Fix}(g)$ ( Counting Theorem )

Two theorems in which $\text{Orb}$ occurs but with a distinct different meaning.

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)