## Related pagesCoordinatizing an affine plane |

We have the following simple theorem.

**Theorem.** The pair
is an affine plane if and only
if
and the following two conditions hold.

- The set S acts sharply 2-transitively on K, i.e. given two pairs
(x
_{1},x_{2}) and (y_{1},y_{2}) of distinct points of K there exists exactly one permuation in S which maps x_{1}to y_{1}and x_{2}to y_{2}, respectively. - Given an element of S, and two points x, y in K with , there exists exactly one such that and

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