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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"

Aron
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    Are you looking for examples or are you looking for a name? – k.stm Jul 28 '19 at 15:01
  • For any action whatsoever of a group $G$ on a set $X$, the quotient group $G/K$ acts faithfully on $X$, where $K$ is the kernel of the action. So your extra condition does not result in any restrictions in the types of space that you get. – Derek Holt Jul 28 '19 at 16:32
  • @DerekHolt I’m reading the question as “are there any topological connections between all faithfully homogeneous spaces for a given group?” in which case I feel like there ought to be some restrictions. The question may very well be, however, “what spaces are faithfully homogeneous spaces for some group,” in which case I agree with your comment. – Santana Afton Jul 28 '19 at 17:17
  • @SantanaAfton, I know that in an homogeneous space is needed only that $G$ acts transitively. Yes, question is what you write what spaces are faithfully homogeneous spaces for some group? 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? – Aron Jul 29 '19 at 09:54
  • @k.stm 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? I find here something – Aron Jul 29 '19 at 10:00
  • Another example I find here, but I can't figure out for a name or for a construction here – Aron Jul 29 '19 at 10:08

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