Theorem 6.13(3) in Casse's book Projective Geometry: An Introduction (Oxford 2006) looks like it might be relevant (it says that in a projective plane the set of “elations” with a given line $\ell$ as axis is an abelian group, and adds that this is known as the “translation group” with axis $\ell$). But I don't know if your definition of an “affine plane” is equivalent to his (a projective plane minus a line), so I don't know if this answers your question. But if you don't know it, you might wish to check the proof.
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