\begin{definition}[Definition:Affine Space/Group Action]
Let $K$ be a field.
Let $\struct {V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set.
Let $\phi: \EE \times V \to \EE$ be a free and transitive group action of $\struct {V, +_V}$ on $\EE$.
Then the ordered pair $\struct {\EE, \phi}$ is an '''affine space'''.
\end{definition}