# Difference between revisions of "Homogeneous principal set"

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− | A '''homogenous principal set''' is a type of [[group]] action on a [[set]]. | + | A '''homogenous principal set''' is a type of [[group]] [[group action|action]] on a [[set]]. |

Let <math>G</math> be a group with a left operation on a set <math>S</math>. The <math>G</math>-group <math>S</math> is called a '''left homogeneous principal set under <math>G</math>''' if it is [[homogeneous set | homogeneous]] (i.e., it has only one [[orbit]]) and for some <math>x\in S</math>, the orbital mapping <math>\alpha \mapsto \alpha x</math> from <math>G</math> to <math>S</math> is [[bijective]]. In this case, ''every'' such mapping is bijective, for if the orbital mapping defined by <math>x_0</math> is bijective, and <math>\alpha_x</math> is the element of <math>G</math> for which <math>\alpha_x x_0 = x</math>, then for any <math>x\in S</math>, the mapping <math>\alpha \mapsto \alpha x = | Let <math>G</math> be a group with a left operation on a set <math>S</math>. The <math>G</math>-group <math>S</math> is called a '''left homogeneous principal set under <math>G</math>''' if it is [[homogeneous set | homogeneous]] (i.e., it has only one [[orbit]]) and for some <math>x\in S</math>, the orbital mapping <math>\alpha \mapsto \alpha x</math> from <math>G</math> to <math>S</math> is [[bijective]]. In this case, ''every'' such mapping is bijective, for if the orbital mapping defined by <math>x_0</math> is bijective, and <math>\alpha_x</math> is the element of <math>G</math> for which <math>\alpha_x x_0 = x</math>, then for any <math>x\in S</math>, the mapping <math>\alpha \mapsto \alpha x = |

## Revision as of 17:50, 9 September 2008

A **homogenous principal set** is a type of group action on a 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 $\alpha \mapsto \alpha \alpha_$ (Error compiling LaTeX. ! Missing { inserted.) 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 .