## Abstract

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}.

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}.

Original language | English |
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Journal | Proceedings of the American Mathematical Society |

Volume | 139 |

Issue number | 2 |

Publication status | Published - 21 Sept 2010 |

## Keywords

- Group Theory
- Topological Algebra
- Infinite Combinatorics