Aspects of infinite permutation groups

Until 1980, there was no such subgroup as ‘infinite permutation groups’, according to the Mathematics Subject Classification: permutation groups were assumed to be finite. There were a few papers, for example [10, 62], and a set of lecture notes by Wielandt [72], from the 1950s. Now, however, there are far more papers on the topic than can possibly be summarised in an article like this one. I shall concentrate on a few topics, following the pattern of my conference lectures: the random graph (a case study); homogeneous relational structures (a powerful construction technique for interesting permutation groups); oligomorphic permutation groups (where the relations with other areas such as logic and combinatorics are clearest, and where a number of interesting enumerative questions arise); and the Urysohn space (another case study). I have preceded this with a short section introducing the language of permutation group theory, and I conclude with briefer accounts of a couple of topics that didn’t make the cut for the lectures (maximal subgroups of the symmetric group, and Jordan groups). I have highlighted a few specific open problems in the text. It will be clear that there are many wide areas needing investigation! 1 Notation and terminology This section contains a few standard definitions concerning permutation groups. I write permutations on the right: that is, if g is a permutation of a set Ω, then the image of α under g is written αg. The symmetric group Sym(Ω) on a set Ω is the group consisting of all permutations of Ω. If Ω is infinite and c is an infinite cardinal number not exceeding Ω, the bounded symmetric group BSymc(Ω) consists of all permutations moving fewer than c points; if c = א0, this is the finitary symmetric group FSym(Ω) consisting of all finitary permutations (moving only finitely many points). The alternating group Alt(Ω) is the group of all even permutations, where a permutation is even if it moves only finitely many points and acts as an even permutation on its support. Assuming the Axiom of Choice, the only nontrivial normal subgroups of Sym(Ω) for an infinite set Ω are the bounded symmetric groups and the alternating group. A permutation group on a set Ω is a subgroup of the symmetric group on Ω. As noted above, we denote the image of α under the permutation g by αg. For the most part, I will be concerned with the case where Ω is countably infinite. Cameron: Infinite permutation groups 2 The permutation group G on Ω is said to be transitive if for any α, β ∈ Ω, there exists g ∈ G with αg = β. For n ≤ |Ω|, we say that G is n-transitive if, in its induced action on the set of all n-tuples of distinct elements of Ω, it is transitive: that is, given two n-tuples (α1, . . . , αn) and (β1, . . . , βn) of distinct elements, there exists g ∈ G with αig = βi for i = 1, . . . , n. In the case when Ω is infinite, we say that G is highly transitive if if is n-transitive for all positive integers n. If a permutation group is not highly transitive, the maximum n for which it is ntransitive is its degree of transitivity. (Of course, the condition of n-transitivity becomes stronger as n increases.) The bounded symmetric groups and the alternating group are all highly transitive. We will see that there are many other highly transitive groups! The subgroup Gα = {g ∈ G : αg = α} of G is the stabiliser of α. Any transitive action of a group G is isomorphic to the action on the set of right cosets of a point stabiliser, acting by right multiplication. The permutation group G on Ω is called semiregular (or free) if the stabiliser of any point of Ω is the trivial subgroup; and G is regular if it is semiregular and transitive. Thus, a regular action of G is isomorphic to the action on itself by right multiplication. A transitive permutation group G on Ω is imprimitive if there is a G-invariant equivalence relation on Ω which is not trivial (that is, not the relation of equality, and not the relation with a single equivalence class Ω). If no such relation exists, then G is primitive. A G-invariant equivalence relation is called a congruence; its equivalence classes are blocks of imprimitivity, and the set of blocks is a system of imprimitivity. A block or system of imprimitivity is non-trivial if the corresponding equivalence relation is. A non-empty subset B of Ω is a block if and only if B∩Bg = B or ∅ for all g ∈ G. A couple of simple results about primitivity: Proposition 1.1 (a) The transitive group G on Ω is primitive if and only if Gα is a maximal proper subgroup of G for some (or every) α ∈ Ω. (b) A 2-transitive group is primitive. (c) The orbits of a normal subgroup of a transitive group G form a system of imprimitivity. Hence, a non-trivial normal subgroup of a transitive group is primitive. In connection with the last part of this result, we say that a transitive permutation group is quasiprimitive if every non-trivial normal subgroup is transitive. Let G be a group, and S a subset of G. The Cayley graph Cay(G,S) is the (directed) graph with vertex set G, having directed edges (g, sg) for all g ∈ G and s ∈ S. If 1 / ∈ S, then this graph has no loops; if s ∈ S ⇒ s−1 ∈ S, then it is an undirected graph (that is, whenever (g, h) is an edge, then also (h, g) is an edge, so we can regard edges as unordered pairs). It is easy to see that Cay(G,S) is connected if and only if S generates G. Most importantly, G, acting on itself by right multiplication, is a group of automorphisms of Cay(G,S). In this situation, G acts regularly on the vertex set of Cay(G,S). Conversely, if a graph Γ admits Cameron: Infinite permutation groups 3 a group G as a group of automorphisms acting regularly on the vertices, then Γ is isomorphic to a Cayley graph for G. (Choose a point α ∈ Ω, and take S to be the set of elements s for which (α, αs) is an edge.)