Orbital

Given a group action of $𝐺$ on $Ω$ , there is a natural group action of $𝐺$ on $Ω \times Ω$ . The orbits of this induced action are called orbitals or 2-orbits, #m/def/group a generalization being $𝑛$ -orbits. The number of orbitals is called the rank.

If $Δ$ is an orbital, then so is its paired orbital

Δ * = {(𝛽, 𝛼) : (𝛼, 𝛽) \in Δ}

and if $Δ = Δ *$ we say $Δ$ is self-paired.

An orbital $Δ \subseteq Ω \times Ω$ admits a natural interpretation as a digraph, known as the orbital digraph, where by abuse of notation we write $V (Δ) = Ω$ and $A (Δ) = Δ$ .

Properties

Suppose $Δ$ is an orbital, and identify it with the corresponding orbital digraph.

$𝐺 \leq A u t (Δ)$ , i.e. $𝐺$ is a subgroup of graph automorphisms.
$Δ$ is arc-transitive — in fact Every arc-transitive digraph is an orbital digraph.
The rank of a transitive group action equals the number of suborbits

#state/tidy | #lang/en | #SemBr