Group action

Orbital

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

If is an orbital, then so is its paired orbital

and if we say is self-paired.

An orbital admits a natural interpretation as a digraph, known as the orbital digraph, where by abuse of notation we write and .

Properties

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

• , i.e. is a subgroup of graph automorphisms.
• 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