# “Affine” vector spaces with groups?

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.