# Finitary symmetric group of countable degree

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

The **finitary symmetric group of countable degree** is defined as the finitary symmetric group on a set of countable size. In other words, it is the subgroup of the symmetric group of countable degree comprising the finitary permutations: the permutations that move only finitely many elements.

## Arithmetic functions

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

order | infinite (countable) | |

exponent | infinite | |

minimum size of generating set | countable | |

subgroup rank | countable |

## Group properties

Property | Satisfied | Explanation |
---|---|---|

finite group | No | |

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

locally finite group | Yes | |

finitely generated group | No | |

periodic group | Yes | |

centerless group | Yes | |

ambivalent group | Yes | |

rational group | Yes | |

monolithic group | Yes | |

one-headed group | Yes | |

almost simple group | Yes |