Homogeneous principal set
Let be a group with a left operation on a set . The -group is called a left homogeneous principal set under if it is homogeneous (i.e., it has only one orbit) and for some , the orbital mapping from to is bijective. In this case, every such mapping is bijective, for if the orbital mapping defined by is bijective, and is the element of for which , then for any , the mapping is the composition of the bijections and ; hence it is a bijection. Thus it is equivalent to say that the operation of on is both free and transitive.
Right homogeneous principle sets are defined similarly.
Examples and Discussion
If is a homogeneous set under an abelian group and operates faithfully on , then is a homogeneous -set. Indeed, suppose are elements of and is an element of for which . Let be any element of , and let be an element of for which . Then
Evidently, the group is a homogeneous set under the left and right actions of a on itself. Sometimes these -sets are denoted and , respectively.
The group of -automorphisms on the left action of on itself () is isomorphic to, and identified with, the set of right translations of , i.e., the opposite group of . Let be a left homogeneous principal -set, and let be an element of . Then the orbital mapping from to is a -set isomorphism. We derive from this isomorphism an isomorphism from the group of -automorphisms of to those of . Note that in general, depends on .