According to Erlangen program, a line is the image of the action on a set $ \mathbf l $ of a subgroup $ G $ of the symmetric group $S_{\mathbf l}$ of $ \mathbf l $, such that given any two pair of distinct points $ (A,B) $ and $ (A^\prime,B^\prime) $ of $ \mathbf l $, there is a unique transformation $ g\in G $ such that $ g(A) = A^\prime $ and $ g(B) = B^\prime $. (In other words, our line is the image of a shaprly $ 2 $-transitive action $ S_{\mathbf l}\geqq G\times\mathbf l\to\mathbf l $, where $ {\leqq} $ is the subgroup relation).
How can one justify this definition?
I read that, given this definition, one can build a set $ \mathbb K $ with all the properties of a field (except that at times distributivity and commutativity of the product may not hold). Such a set, apparently, "makes it possible to assign absicssae, in a given frame of reference, to the points of $ \mathbf l $, and by identifying the points of the line with their abscissae, affine mappings are all the mappings of the form $ ax + b $, where $ a\neq 0 $.
How is $ \mathbb K $ constructed?