A characterization of some \(2\)-transitive groups (Q1325073)

The author extends part of the classification of the so-called finite ``Zassenhaus group'' to the infinite case. He considers the following hypotheses on a permutation group \(G\) acting on \(\Omega\): (H1) \(G\) is 2- transitive, \(H\) the stabilizer of a point in \(\Omega\) and \(t\) an involution in \(G \setminus H\), \(D = H \cap H^ t\) and \(Q\) is a normal complement of \(D\) in \(H\). (H2) \(Q\) is abelian, and if \(d\) is an involution in \(D\) then \(C_ Q(d) = 1\). (H3) For \(d \in D\) the equality \(d^ t = d^{-1}\) holds; thus \(D\) is abelian. The main result is the following Theorem: If (H1), (H2) and (H3) are satisfied, then one of the following assertions holds: (I) \(G\) is similar to the affine group \(F \rtimes F^ \#\) of some commutative field \(F\). (II) \(\langle Q^ x \mid x \in G\rangle \cong PSL(2,K)\) for some commutative field \(K\) and \(G\) is similar to a group \(G_ 1\) such that \(PSL(2,K) \leq G_ 1 \leq PGL(2,K)\). (III) \(Q\) has exponent 2 and there exists a mapping \(h : Q^ \# \to D\) such that \(h(x^ d) = d^ 2 h(x)\) for \(x \in Q^ \#\), \(d\in D\); \(x^{h(xy)}=x^{h(x)} x^{h(y)}\) for \(x,y \in Q^ \#\), \(x \neq y\); \((th(x)x)^ 3 = 1\) for \(x \in Q^ \#\). Here \(F^ \# = F\setminus \{0\}\) and \(Q^ \# = Q \setminus \{1\}\). Furthermore two corollaries are derived which give characterizations of the groups \(PGL(2,K)\): 1) If in case III of the Theorem in addition \(h(y)h(x)^ - \in D^ 2\) for all \(x,y\in Q^ \#\) (which holds in particular if \(D^ 2 = D\) or if \(D\) has finite exponent) then \(G\) is similar to \(PGL(2,K)\) for some commutative field \(K\) of characteristic 2. 2) If \(G\) is sharply 3-transitive and \(| \Omega| \geq 3\), and if any element of \(G\) exchanging two different points is an involution then \(G\) is similar to \(PGL(2,K)\) for some commutative field \(K\). (This result had been obtained previously by \textit{J. Tits} [in Acad. R. Belg., Bull. Cl. Sci. 35, 197-208, 224-233, 568-589, 756-773 (1949; Zbl 0034.305, Zbl 0035.296, and Zbl 0036.295)]).