Infinite permutation groups II. Subgroups of small index

Abstract

It was shown by Dixon, Neumann, and Thomas [3] that if 52 is a countably infinite set, and H is a subgroup of the full symmetric group Sym(SZ) on Q of index less than 2no, then H contains the pointwise stabilizer in Sym(Q) of a finite subset of 52. Referring to the analogous statement for an arbitrary group G (which will usually be a subgroup of Sym(Q) for some 52 with ISZJ = N,) as the “conjecture on subgroups of small index for G” it was shown by Evans [S] that the conjecture holds for the general linear group of a vector space of dimension K, over a countable (or finite) division ring, and Bruyns [Z] obtained partial information about the conjecture in the case where G = Aut Q, the group of homeomorphisms of Q to itself. The conjecture was formulated in [3] for two particularly interesting cases, Aut Q and A(Q) (the latter being the group of order-preserving permutations of Q). Macpherson [S] was more ambitious and suggested that the conjecture should hold for the automorphism group of any NOcategorical structure (though he weakened the hypothesis IG : HI < 2’O to IG : HI < K,), and Evans [6] showed how to reformulate the conjecture using ideas about “closed” subgroups of G. Recently Hrushovski has found a counter-example to the general case of the conjecture, though it still seems likely that it will hold in a wide class of instances. The object here is to establish the truth of the conjecture in the two main cases raised in [3]. We shall prove it in some other cases too, principally the group of automorphisms of the countable atomless Boolean algebra. This is isomorphic as a graq to Aut C, where C is the Cantor discontinuum, though not as a permutation group (in fact Aut C has degree 2’“). Thus although we work mainly with Aut C (or rather Aut 2”) the statement of the result comes out rather differently when expressed in this group. Namely it says: if H is a subgroup of Aut C of index less than 2X0 494 0021-8693/89 $3.00