By the very definition of a nearfield, the affine group of K acts sharply 2-transitive on K.

We may also think of K as a vector space over its kernel F:=Ker(K). Then the set

is a subgroup of the linear group of K over F acting sharply transitive
on
Here, l_{a} denotes the multiplication by a from the left, i.e.
l_{a}(x)=ax. In fact, G is just the
spreadset associated to K
minus the null. Conversely, given a finite dimensional vector space
V over a skewfield F and a subgroup G of GL(V) which is sharply transitive
on the vectors distinct from zero, then G arises from some
nearfield as described above.

In the finite case, each sharply 2-transitive action is induced by a nearfield. This is an immediate consequence of the Frobenius theorem, for example. In particular, the classification of finite nearfields implies a classification of these permutation groups.

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