Homogeneous principal set

A homogenous principal set is a type of group action on a set.

Let $G$ be a group with a left operation on a set $S$. The $G$-group $S$ is called a left homogeneous principal set under $G$ if it is homogeneous (i.e., it has only one orbit) and for some $x\in S$, the orbital mapping $\alpha \mapsto \alpha x$ from $G$ to $S$ is bijective. In this case, every such mapping is bijective, for if the orbital mapping defined by $x_0$ is bijective, and $\alpha_x$ is the element of $G$ for which $\alpha_x x_0 = x$, then for any $x\in S$, the mapping $\alpha \mapsto \alpha x = \alpha \alpha_x x_0$ is the composition of the bijections $\alpha \mapsto \alpha \alpha_x$ and $\alpha \mapsto \alpha x_0$; hence it is a bijection. Thus it is equivalent to say that the operation of $G$ on $S$ is both free and transitive.

Right homogeneous principle sets are defined similarly.

Examples and Discussion

If $S$ is a homogeneous set under an abelian group $G$ and $G$ operates faithfully on $S$, then $E$ is a homogeneous $G$-set. Indeed, suppose $f,g$ are elements of $G$ and $x$ is an element of $S$ for which $fx \neq gx$. Let $y$ be any element of $S$, and let $\alpha_y$ be an element of $G$ for which $\alpha_y x = y$. Then \[fy = f\alpha_y x = \alpha_y fx \neq \alpha_y gx = g \alpha_y x = gy.\]

Evidently, the group $G$ is a homogeneous set under the left and right actions of a $G$ on itself. Sometimes these $G$-sets are denoted $G_s$ and $G_d$, respectively.

The group of $G$-automorphisms on the left action of $G$ on itself ($G_s$) is isomorphic to, and identified with, the set of right translations of $G$, i.e., the opposite group $G^0$ of $G$. Let $S$ be a left homogeneous principal $G$-set, and let $x$ be an element of $S$. Then the orbital mapping $\omega_x : \alpha \mapsto \alpha x$ from $G$ to $S$ is a $G$-set isomorphism. We derive from this isomorphism an isomorphism $\psi_x$ from the group of $G$-automorphisms of $G_s$ to those of $S$. Note that in general, $\psi_x$ depends on $x$.

See also

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