Normal structure of the one-point stabilizer of a doubly-transitive permutation group. II

The main result is that the socle of the point stabilizer of a doubly-transitive permutation group is abelian or the direct product of an abelian group and a simple group. Under certain circumstances, it is proved that the lengths of the orbits of a normal subgroup of the one point stabilizer bound the degree of the group. As a corollary, a fixed nonabelian simple group occurs as a factor of the socle of the one point stabilizer of at most finitely many doubly-transitive groups. Introduction. This paper is a continuation of our study of the normal structure of the one-point stabilizer of a doubly-transitive permutation group, a study which we began in part I [9]. Here we prove the following theorems: THEOREM A. Let G be a doubly-transitive permutation group on a finite set X. Suppose that x is an element of X. Then either (i) Gx has an abelian normal subgroup # 1, or (ii) G. has a unique minimal normal subgroup, and this minimal normal subgroup is simple. Actually, we prove slightly more than this. In fact, it is shown that the socle of Gx is either an abelian group or the direct product of an abelian group (possibly 1) and a simple group. Using this and a recent theorem of Aschbacher [11], it follows that in an unknown doubly-transitive group of minimal degree the socle of G. is either abelian or simple. THEOREM B. Let G be a doubly-transitive permutation group on a finite set X and x an element of X. Suppose NX is a normal subgroup of G,. Let [A = n and suppose that on X x all orbits of NX have length s. Then either (i) NX is semiregular on X x, or (ii) G is a normal extension of Ln(q) (in its natural doubly-transitive representation), or (iii) n < 2(s 1)2. Presented to the Society, January 26, 1973; received by the editors May 6, 1974. AMS (MOS) subject classifications (1970). Primary 20B20; Secondary 20B10, 20B25.

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