a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively
But I try understand What kind of space I get if I add also faithfully condition not only a transitive (group) action.
Example: I know that an affine space is a group action on a set, specifically a vector space acting on a set faithfully and transitively, so I search for a name for this algebraic structure between affine space and vector space. Exist a name or a way to build algebraically this space?
For vector space I look here
And I find also something here
All depends the point of view you like (at that is a matter of taste), IMHO, the "most connected" (with the rest of mathematics) is that of an orbit under translations. Then you have (a) a group of translations $(T_v)_{v\in V}$ (the additive group $(V,+)$) on a set $A$ , but (b) there must be only one orbit (i.e. the action must be transitive and (c) faithful. These structures, very important in mathematics, are called principal homogeneous space or torsors and can be thought as "the group (here $V,+$) without priviledged origin or unit"