# Fundamental group of Klein bottle

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## Definition

The **fundamental group of Klein bottle** is defined in the following equivalent ways:

- It is the fundamental group of the Klein bottle.
- It is the Baumslag-Solitar group .
- It is given by the presentation:

Note that the group admits the infinite dihedral group as a quotient group.

## Group properties

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

finitely generated group | Yes | See presentation |

finitely presented group | Yes | See presentation |

one-relator group | Yes | See presentation |

solvable group | Yes | |

metacyclic group | Yes | |

polycyclic group | Yes | |

supersolvable group | Yes | |

Noetherian group | Yes | Follows from being polycyclic |

metabelian group | Yes | |

solvable group | Yes | |

residually nilpotent group | Yes |