Subgroups of small index in infinite symmetric groups. II

Throughout this paper Q will denote an infinite set, S:= Sym(Q) and G is a subgroup of S. If n: = |Q|, the cardinal of Q, then |.S| = 2". Working in ZFC, set theory with Axiom of Choice (AC), we shall be seeking the subgroups G with \S: G\ < 2". If A £ Q then S^ (respectively G{A}) denotes the setwise stabiliser of A in S (respectively in G); 5(A) and G(A) denote pointwise stabilisers; we identify S(A) with Sym(fi —A). A subset £ of Q such that |Z| = |Q — 1 | = |O| is known as a moiety ofQ. Suppose now that |Q| = n = Ko. If there is a finite subset A of Q such that 5(A) ^ G then certainly \S:G\ ^ Xo. Our theme is a rather strong converse: