Definition:Affine Space/Group Action

Definition
Let $K$ be a field.

Let $\left({V, +_V, \circ}\right)$ be a vector space over $K$.

Let $\phi: \struct{V, +_V} \times \mathcal E \to \mathcal E$ be a free and transitive group action of $\struct{V, +_V}$ on $\mathcal E$.

Then the ordered pair $\tuple{\mathcal E, \phi}$ is an affine space.

Also see

 * Equivalence of Definitions of Affine Space