Generating the Infinite Symmetric Group Using a Closed Subgroup and the Least Number of Other Elements

Let S∞ denote the symmetric group on the natural numbers N. Then S∞ is a Polish group with the topology inherited from NN with the product topology and the discrete topology on N. Let d denote the least cardinality of a dominating family for NN and let c denote the continuum. Using theorems of Galvin, and Bergman and Shelah we prove that if G is any subgroup of S∞ that is closed in the above topology and H is a subset of S∞ with least cardinality such that G ∪H generates S∞, then, |H| ∈ {0, 1, d, c}. The symmetric group S∞ is a Polish group under the topology inherited from the product topology on NN with the discrete topology on N; see [8, Section 9.B(7)] for further details. We will refer to subgroups G of S∞ that are closed in this topology as closed subgroups. It is a well-known fact that the closed subgroups of S∞ are precisely the automorphism groups of relational structures on N; see, for example, [4, Theorem 5.8]. Such automorphism groups have been widely investigated; see for example [9] and the references therein. The theorem of Bergman and Shelah from [3] on which our main theorem (Theorem 1.3) is based involves the following equivalence relation ≈ on subgroups of S∞. Let G and H be (not necessarily closed) subgroups of S∞. Then G ≈ H if there exists a countable C ⊆ S∞ such that the subgroup 〈H,C 〉 generated by H and C equals 〈G,C 〉. Throughout, we write functions to the left of their argument and compose from right to left. The following two subgroups of S∞ are representatives of two of the classes under ≈. Let A be a partition of N into sets A1, A2, . . . where |Ai| = i for all i ≥ 1 and let B be a partition of N into sets B0, B1, . . . with |Bi| = 2 for all i ∈ N. Then define HN = { f ∈ S∞ : f(Ai) = Ai for all i ≥ 1 } H2 = { f ∈ S∞ : f(Bi) = Bi for all i ∈ N }. It is straightforward to verify that HN and H2 are closed subgroups of S∞. If G is a subgroup of S∞ and Σ ⊆ N, then the pointwise stabilizer of Σ in G is the subgroup G(Σ) = {f ∈ G : f(σ) = σ for all σ ∈ Σ}. We require the following slightly weaker version of the main theorem in Bergman and Shelah [3]. 1991 Mathematics Subject Classification. Primary 20B07. Secondary 20B07, 54H11. c ©XXXX American Mathematical Society