Principal Homogeneous Space

Free: if g ⋅ p = p g\cdot p=p g ⋅ p = p for some p ∈ P p\in P p ∈ P , then g = e g=e g = e .
Transitive: for any p , q ∈ P p,q\in P p , q ∈ P , there exists g ∈ G g\in G g ∈ G with g ⋅ p = q g\cdot p=q g ⋅ p = q .

Choosing a basepoint identifies it with the group

Fix $p_0\in P$ . The map

\theta_{p_0}:G\to P,\qquad \theta_{p_0}(g)=g\cdot p_0

is a diffeomorphism. This identifies $P$ with $G$ , but the identification depends on the choice of $p_0$ , so there is generally no canonical “origin” in a torsor.

Examples and related notions

$G$ acting on itself by left translation is a principal homogeneous space.
More generally, a transitive action with stabilizer $H$ gives a homogeneous space $G/H$ ; compare quotients. A principal homogeneous space is the special case $H=\{e\}$ .

Principal homogeneous spaces are the geometric backdrop for principal bundles and symmetry without a preferred identity element.

Equivalent characterizations

Equivalently, the action is simply transitive: for each $p,q\in P$ there is a unique $g\in G$ with $g\cdot p=q$ .