The Non-Solvable Rank 3 Affine Planes

Large doubly transitive orbits on a line

“…Hence either n = 4 or n = 5. If II = PG(2, 4) then any subgroup G of /TL(2,4) isomorphic to D 6 = AGL(\, 3) and containing the involution induced by the Frobenius automorphism of GF(4) fixes two points on / and acts 2-transitively on the remaining ones. Thus (1).…”

“…Thus (1). If II = PG (2,5) the group SL(2, 5) induces A 5 on / and any G = D 6 inside A 5 has two 2-transitive point-orbits on / both of length 3, and hence (2).…”

“…Apart from a few numerical values of n, Hiramine shows; that the socle of G, where G denotes the group induced by G on 0, is an elementary abelian p-group for some prime p, the plane II has order n = p r and either G o < TL(1, p r ) orSL(2, p r ) <G 0 < TL (2, p r ). In 1999, Biliotti, Jha and Johnson classified the translation planes II for v = n, n ^ 2 6 , when / is the line at infinity and G < ATHX, p r ). In 2000, Ganley, Jha and Johnson [20] classified the triple (n, 0, Cj) for v=n, when n is a translation plane, / is an affine line and G is non solvable.…”

“…In particular, for the finite groups admitting a 2-transitive permutation representation we have the following classification. 5 and G o /R < S 5 ; (h) G o = SL (2,13), p d = 3 6 .…”

“…Note that a classification of the projective extensions of translation planes, when / is an affine line and one of the situations (1) or (3) of Hiramine's theorem occurs, is not available. Nevertheless, there are several examples corresponding to each of these situations (see [6]). In particular, in the examples referring to the situation (3), A 5 is involved in G o in many cases.…”