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.

Orbitals

6 days ago

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