✆ ✄ ✂ ✂ ✂ ✞ ✄ ✁ ✂ ✝ ✄ ✁ Connection with model theory A countable first-order structure is countably categorical if it is the unique countable model of its Orbits on n -tuples first-order theory. Theorem M is countably categorical if and only if � M Aut ✁ is oligomorphic. Peter J. Cameron If so, then the number of n -types of the theory of M is Queen Mary, University of London n � n ∑ F S ✆ k ✁ F k n p.j.cameron@qmw.ac.uk ☎ 1 k � n ✁ are the Stirling numbers ✆ k where the coefficients S Computation, Models and Groups of the second kind. Cambridge, 11-12 January 2001 Moreover, every oligomorphic group of countable degree which is closed (in the topology of pointwise convergence) is the automorphism group of a countably categorical structure. 1 3 Introduction Generating function � x Let G be a permutation group on a set Ω (usually Let F ✁ be the exponential generating function of � F n infinite). Then G is oligomorphic if, for each natural ✁ , the formal power series number n , the number F n of G -orbits on n -tuples of ∞ F n x n � x distinct elements of Ω is finite. ∑ F n ! ☎ 0 n In general this is only a formal power series, but of The main question is: what can be said about the growth of the sequence F n ? course we can ask: when does it converge in some neighbourhood of the origin? This question may be easier than the corresponding � e x � x � x 1 Note that F F ✁ , where F question about the number of orbits of G on ✁ is the n -element subsets, but has been less studied. In exponential generating function for the sequence � F ✁ , so one series converges if and only if the other terms of combinatorial enumeration, it corresponds to n does. counting labelled, rather than unlabelled, structures. 2 4
✆ ✄ ✁ ✏ ☛ ✝ ✄ ✁ ✆ ✚ ✝ ✄ ✠ ✘ ✁ � ✄ � ✄ ✁ ✑ ✄ ✄ ✁ ✞ ✄ ☛ ✞ ✁ � ✙ ✔ � ✄ ✑ ✆ Digression: cycle index Finite groups The result of Boston et al. is in fact a special case of Suppose that Ω is finite. Boston et al. proved: an older result, which considers cycles of arbitrary length instead of just fixed points. Theorem The probability generating function for the number of fixed points of a random element of G is � x The probability generating function for the numbers 1 F ✁ . of cycles of all possible lengths is the classical cycle index of the permutation group, as developed by (The p.g.f. is the polynomial ∑ p i x i , where p i is the Redfield, P´ olya, de Bruijn and others. It is a proportion of elements in G having exactly i fixed polynomial in indeterminates s 1 ✆ s 2 ✝✗✝✗✝ , where s i points.) 1 for all i 1 , records cycles of length i . If we set s i we obtain the p.g.f. for fixed points. In particular, the proportion of fixed-point-free ✞ 1 elements in G is F ✁ . Now we have three (actual or potential) generalisations of the result: to oligomorphic groups; ✞ 1 For some oligomorphic groups, F ✁ is defined (e.g. to linear groups; and to cycles of arbitrary length. by analytic continuation), but it is not clear whether The possibility of combining two (or all three) of these any meaning can be assigned to it. generalisations exists, but has not been fully realised. 5 7 Digression: linear groups Macpherson’s Theorem The result of Boston et al. can be written as A permutation group is primitive if it preserves no ✟ j n F i non-trivial equivalence relation. (Note that in the ∑ ✠ p j i ! i countably categorical case, an invariant equivalence j ☎ i relation would be definable without parameters.) A which can be inverted to give permutation group is highly transitive if F n 1 for all ✟ j � x n F j e x converges everywhere. ✞ 1 n , so that F ∑ ✁ j ✡ i p i j ! i j ☎ i � q Dugald Macpherson proved: There is an analogue for linear groups over GF ✁ . We replace i ! (the order of the symmetric group) by ☛ GL � i ☛ , and Theorem If G is oligomorphic and primitive but not ☞ j ✍ j ✆ q ✌ by the Gaussian coefficient ✎ q ; i i also, let L i be the number of orbits of a linear group highly transitive, then � n GL ✁ on linearly independent i -tuples, and P i G ✆ q n ! F n � n the proportion of elements of G whose fixed point set p � x is precisely an i -dimensional subspace. Then for some polynomial p . In particular, F ✁ has radius n L i j of convergence at most 1 . ∑ ☛ GL � i P j ✆ q i ✒ q j ☎ i For example, the group of order-preserving which can be inverted to give n ! for permutations of the rational numbers has F n � 1 n � x L j j ✞ 1 ∑ ✡ 1 ✔✖✕ 2 ✁ j ✡ i q ✓ j ✡ i ✓ j ✡ i ☛ GL all n , so that F 1 x � j P i ✁ , with radius of i ✆ q ✒ q convergence 1 . j ☎ i 6 8
✙ ✞ ✄ ✄ ✞ ✞ ✄ ✄ ✄ ✁ ✛ ✞ ✘ Groups with slow growth Highly homogeneous groups Thus for primitive groups other than the highly homogeneous ones, the slowest possible growth rate A permutation group is highly homogeneous if it is an exponential times a polynomial. permutes transitively the set of all n -element subsets of Ω , for all natural numbers n . I proved the following: Problem: What can be said about those groups for � x which F ✁ nas non-zero radius of convergence (that Theorem If G is highly homogeneous but not highly is, F n grows no faster than c n n ! for some c ? transitive, then there is a linear or circular order preserved or reversed by G . Empirically, the known examples come from either � n � n “circular” or “treelike” objects, or combinations of n ! , n ! ✚ 2 , 1 ✁ ! , and 1 ✁ ! ✚ 2 in the We have F n these.There appears to be a big overlap with the four cases (for large enough n ). primitive Jordan groups classified by Adeleke, Macpherson and Neumann. 9 11 Merola’s Theorem Examples Example 1 Take a dense subset of the set of Francesca Merola recently strengthened ✞ x for Macpherson’s Theorem as follows. complex roots of unity containing one of x and � y all x . An arc joins x to y if 0 arg π . We have ✚ x � 2 n ✁✜✛ Theorem There is a constant c 1 such that, if G is F n 2 ✁ !! . primitive but not highly homogeneous, then Example 2 Take the amalgamation class of c n n ! � n F n p endvertex structures of boron trees (trees in which each vertex has valency 1 or 3 ). We have for some polynomial p . In particular, the radius of � 2 n � x F n 5 ✁ !! . ✁ is at most 1 convergence of F ✚ c . Example 3 Embed the trees of Example 2 in the 1 ✝ 174 Her proof gives c ✝✗✝✗✝ , but it is conjectured that 2 (this would be best the result holds with c plane. The leaves are then circularly ordered, and we may impose the structure of Example 1 on them. possible). 10 12
✣ ✢ ✁ ✢ ✘ ✢ ✣ ✙ ✞ ✢ ✙ ✄ ✄ ✄ ✏ Outline of the proof, II The other important ingredient is the following. If we Smoothness � n can find f ✁ sets of size n in different G -orbits, such that the group of permutations induced on each set The numbers F n should grow not only rapidly but also � n by its setwise stabiliser has order at most g ✁ , then smoothly in general. It is hard to formulate a general we have � n problem which is informative for all oligomorphic ✁ n ! f � n F n groups. For groups of small growth, we can ask the g ✁✤✝ following question. � n � n � F n So the trick is to choose f ✁ and g ✁ so that ✁ 1 ✕ n tends to a limit? Is Problem: Is it true that ✚ n ! � n � n f ✁✗✚ g ✁ grows exponentially. This requires some it even true that F n ✚ nF n ✡ 1 tends to a limit? If so, what fine tuning! are the possible values of the limit? The proof involves doing this for a variety of different combinatorial structures. 13 15 Outline of the proof, I Beginning the induction Merola’s proof follows the main steps of Macpherson’s. Case k 2 : If G is not 2 -homogeneous, then it is a � c group of automorphisms of a graph. If G is Let ✁ be the class of oligomorphic groups which 2 -homogeneous but not 2 -transitive, then it is a group satisfy of automorphisms of a tournament. c n n ! � n F n p Case k 3 : If G is not 3 -homogeneous, it is a group for some polynomial p . Note that a transitive group � c of automorphisms of one of a variety of combinatorial belongs to ✁ if and only if the point stabiliser does. structures including Steiner systems, 2 -trees, etc. If G is 3 -homogeneous but not 2 -primitive, it preserves We prove, by induction on k , that (for a suitable a treelike structure. If G is 2 -primitive then we can 1 ) a primitive but not highly constant c � c apply the arguments of the preceding case to G α . ✁ is k -transitive. homogeneous group in For each class of structures we have to choose c The induction step from k to k 1 is trivial if k 3 . If G � c sufficiently small for the proof to work. The smallest ✁ , then G is k -transitive, so is a primitive group in � k 1 ✝ 174 value c ✝✗✝✗✝ occurs in the case of tournaments. 1 G α is ✁ -transitive (and hence primitive), hence � k 1 G α is k -transitive, hence G is ✁ -transitive. 14 16
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