Sharply 2-transitive sets of permutations and groups of affine projectivities

Using new results on sharply transitive subsets, we determine the groups of projectivities of finite affine planes, apart from (unknown) planes of order 23 or 24. The group of all projectivities of a geometry G is a measure for the complexity of G: this group tends to be rather large if G is far from being a classical geometry. See [PS81] for more information on the role of projectivities in geometry. In Section 1 we consider almost simple finite permutation groups which contain a sharply 2-transitive subset. The results of this section yield Theorem 2.3 on affine projectivities of finite affine planes that are not translation planes. 1 Sharply transitive subsets Let Ω be a set. A set S of permutations of Ω is said to be sharply transitive if for all α, β ∈ Ω there is exactly one s ∈ S with α = β. If t is any permutation of Ω, then S is sharply transitive if and only if this holds for tS. Thus, if we study the (non-) existence of sharply transitive sets S of permutations, we may assume that 1 ∈ S. Then all elements in S \ {1} are fixed-point-free. Let Ω be the set of pairs of distinct elements from Ω. Then S is said to be sharply 2-transitive on Ω if S is sharply transitive on Ω. If this holds, then each stabilizer Sω = { s ∈ S | ω = ω } is sharply transitive on Ω\{ω}. Some 2-transitive groups G do not contain any sharply 2-transitive subset for the simple reason that a point-stabilizer Gω does not contain a subset which is sharply transitive on Ω\{ω}. Examples for this are