# Symmetric group of countable degree

From Groupprops

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## Definition

The **symmetric group of countable degree** is defined as the symmetric group on a set of countable size. This group is denoted or .

The typical countable set used is , the set of natural numbers.

## Group properties

Function | Value | Explanation |
---|---|---|

abelian group | No | |

centerless group | Yes | symmetric groups are centerless |

nilpotent group | No | |

solvable group | No | |

finite group | No | |

finitely generated group | No | |

complete group | Yes | symmetric groups on infinite sets are complete |

ambivalent group | Yes | |

rational group | Yes |