c © TÜB_ITAK Splitting of Sharply 2-Transitive Groups of Characteristic 3

We give a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3. Key Words: Sharply 2-transitive groups, Permutation groups. A sharply 2-transitive group is a pair (G;X), where G is a group acting on the set X in such a way that for all x; y; z; t 2 X such that x 6 = y and z 6 = t there is a unique g 2 G for which gx = z and gy = t: From now on, (G;X) will stand for a sharply 2-transitive group with jXj 3. We x an element x 2 X. We let H: = fg 2 G: gx = xg denote the stabilizer of x. Finally we let I denote the set of involutions (elements of order 2) of G. It follows easily from the denition that the group G has an involution; in fact any element of G that sends a distinct pair (y; z) of X to the pair (z; y) is an involution by sharp transitivity. It is also known that I is one conjugacy class and the nontrivial elements of I2 cannot x any point (See Lemma 1 and Lemma 4). Then one can see that I2 cannot have an involution if H has an involution. In case H has no involution, one says that char(G) = 2. Let us assume that char(G) 6 = 2. Then I2 n f1g is one conjugacy class [1, Lemma 11.45]. Since I2 is closed under power taking, either the nontrivial elements of I2 all have order p for some prime p 6 = 2 or I2 has no nontrivial torsion element. One writes char(G) = p or char(G) = 0 depending on the case. One says that G splits if the one point stabilizer H has a normal complement in G. It is not known whether or not an innite sharply 2-transitive group splits, except for thos