Examples
Every group G can itself be thought of as a left or right G-torsor under the natural action of left or right multiplication.
Another example is the affine space concept: the idea of the affine space A underlying a vector space V can be said succinctly by saying that A is principal homogeneous space for V acting as the additive group of translations.
The flags of any regular polytope form a torsor for its symmetry group.
Given a vector space V we can take G to be the general linear group GL(V), and X to be the set of all (ordered) bases of V. Then G acts on X in the way that it acts on vectors of V; and it acts transitively since any basis can be transformed via G to any other. What is more, a linear transformation fixing each vector of a basis will fix all v in V, hence being the neutral element of the general linear group GL(V) : so that X is indeed a principal homogeneous space. One way to follow basis-dependence in a linear algebra argument is to track variables x in X. Similarly, the space of orthonormal bases (the Stiefel manifold of n-frames) is a principal homogeneous space for the orthogonal group.
In category theory, if two objects X and Y are isomorphic, then the isomorphisms between them, Iso(X,Y), form a torsor for the automorphism group of X, Aut(X), and likewise for Aut(Y); a choice of isomorphism between the objects gives an isomorphism between these groups and identifies the torsor with these two groups, and giving the torsor a group structure (as it is a base point).
Read more about this topic: Principal Homogeneous Space
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