Example

(1) Let G operate on itself by left (resp. right) translation. Thus a left (resp. right) homogeneous principal G-set structure is defined on G, which is sometimes denoted by $G_s$ (resp. $G_d$).

(2) Let E be a homogeneous set under a commutative group G. If G operates faithfully on E, the latter is a homogeneous principal G-set.

(3) Let E and F be two isomorphic sets with structures of the same species and let Isom(E, F) be the set of isomorphisms of E onto F (with the given structures). The group Aut(E) of automorphisms of E (with the given structure) operates on Isom(E, F) on the right by the law $(\sigma, f) \mapsto f \circ \sigma$ and Isom(E, F) is a right homogeneous principal Aut(E)-set. Similarly, the group Aut(F) operates on Isom(E, F) on the left by the law $(\sigma, f) \mapsto \sigma \circ f$ and Isom(E, F) is a left homogeneous principal Aut(F)-set.

(4) *A homogeneous principal set under the additive group of a vector space is called an affine space (cf. II, § 9, no. 1).*

The group of automorphisms of the homogeneous principal G-set $G_s$ (Example 1) is the group of right translations of G which is identified with $G^0$ (no. 5, Proposition 5). Let E be a homogeneous principal G-set and $a$ an element of E. The orbital mapping $\omega_a$ defined by $a$ is an isomorphism of the G-set $G_s$ onto E. By transporting the structure an isomorphism $\psi_a$ is derived of $G^0$ onto Aut(E). It should be noted that $\psi_a$ in general depends on a; more precisely, for $\alpha \in G$,

For, writing $\delta_a$ for the translation $x \mapsto x \alpha$ on G,

and whence