Sharply 2-transitive groups

Let K be a nearfield. The affine group of K is defined to be the set of all transformations with i.e.

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, la denotes the multiplication by a from the left, i.e. la(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.

Contributed by Hauke Klein
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