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I wish to consider a structure that is like an affine space, but does not use a vector space as the affine structure, rather uses a group. That is, we shall "forget" the scaling structure of the vector space, while still keeping the group structure:

I define an algebraic structure AffineGroup $(G, S, -, \curvearrowright)$ where:

  • $S$ is any set of elements
  • $G \equiv (G_{set}, e, *)$ is a group
  • $-: S \times S \rightarrow G_{set}$ is a "distance function"
  • $\curvearrowright : G_{set} \times S \rightarrow S$ is a group action of $G$ on $S$.
  • $\forall s \in S, (s - s) = e$
  • $ \forall s_1, s_2 \in S,~ (s_2 - s_1) \curvearrowright s_1 = s_2$

Has such a structure been studied in the literature? (I feel it must have been). What is this structure called, and where can I look for more about this?

Also, bonus question: Must $G$ be abelian? Can we consider a non-commutative group (unlike the vector space case, where we needed to have $(V, +)$ be abelian.

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Thanks to @Eic Wofsey for the link!

It appears that this algebraic structure is called as a Torsor.

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