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}.