tailieunhanh - Splitting of sharply 2-transitive groups of characteristic 3

We give a much simpler proof of this fact, in fact an experienced reader can directly go to the proof the Theorem, which contains only a simple computation. We give a group theoretic proof of the splitting of sharply 2-transitive groups of characteristic 3. | Turk J Math 28 (2004) , 295 – 298. ¨ ITAK ˙ c TUB Splitting of Sharply 2-Transitive Groups of Characteristic 3 Seyfi T¨ urkelli Abstract 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 ∈ X such that x 6= y and z 6= t there is a unique g ∈ G for which gx = z and gy = t. From now on, (G, X) will stand for a sharply 2-transitive group with |X| ≥ 3. We fix an element x ∈ X. We let H := {g ∈ G : gx = x} 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 definition 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 I 2 cannot fix any point (See Lemma 1 and Lemma 4). Then one can see that I 2 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 I 2 \ {1} is one conjugacy class [1, Lemma ]. Since I 2 is closed under power taking, either the nontrivial elements of I 2 all have order p for some prime p 6= 2 or I 2 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 infinite sharply 2-transitive group splits, except for those 295 ¨ TURKELL I˙ of characteristic 3. Results in this direction for some special cases can be found in [1, §] and [2, ch 2]. We will prove that if char(G) = 3 then G splits, a result of W. Kerby [2, Theorem ]. But Kerby’s proof is in the language of near domains and is not easily accessible. .