The Classification of Some Infinite Jordan Groups

WHEN we speak of a 'back and forth' construction for proving the isomorphism of two countable structures we call to mind the famous theorem of Cantor that any countable dense linearly ordered set without end-points is order-isomorphic to Q. Cantor's own proof requires only the 'going forth' part of the construction, however, and Adrian Mathias, on noticing this, asked for a classification of those Xo-categorical theories (or relational structures) for which 'forth always suffices'. In the hands of Peter Cameron (see [5], pp. 124-129) Mathias's problem has led to intriguing questions about permutation groups. Let ft be a set, G a subgroup of Sym (ft). Recall that a Jordan set (for G in ft) is a subset Z of ft such that |2| > 1 and the pointwise stabiliser G(n-x) of its complement is transitive on 2. By a J-set we shall mean a set which is either a Jordan set or a singleton and, later in this paper, by a proper J-set for (G, ft) we will mean a J-set different from ft (warning: the adjective 'proper' applied to Jordan sets has a slightly more sophisticated meaning—see [3]). One particularly well-focused question that Cameron posed in relation to the sufficiency of going forth (see [5], pp. 128) was: