<i>p</i>-solvable doubly transitive permutation groups

Abstract

In this paper doubly transitive and 3/2-transitive permutation groups are classified under hypotheses somewhat weaker than solvability. We mention two examples. Let S^(p n ) denote the group of semilinear transformations over GF(p n ). The following combines a result of Huppert on solvable 2-transitive groups and a result of Zassenhaus on sharply 2-transitive groups. THEOREM I. Let be a p-solvable doubly transitive permutation group with O p () <1>. Then deg = p n for some n and we have one of the following: (i) < c S'Xv n ), () is solvable and p = 3 2 , 5 2 , 7 2 , II 2 , 23 2 or 3 4 , or (iii) is nonsolvable and p n = II 2 ,19 2 , 29 2 or 59 2 . The second result reads better as a theorem on linear groups. THEOREM II. Let group act faithfully on vector space $ over GF(p) and let act 1/2-transitively but not semiregularly on W. If is imprimitive as a linear group, then is solvable and we have one of the following: (i) | 33 | = p2n f or p 2 an d is a specific group of order A(p n -1), (ii) I 53 I = 3 4 and | | = 32, or (iii) | 53 | = 2 6 and | | = 18. 556 D. S. PASSMAN