# Baumslag-Solitar group:BS(1,2)

From Groupprops

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## Definition

This group is defined as the Baumslag-Solitar group with parameters 1 and 2, i.e., the group . Explicitly, it is defined by means of the following presentation:

## Group properties

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

finitely generated group | Yes | See presentation above |

finitely presented group | Yes | See presentation above |

one-relator group | Yes | See presentation above |

word-hyperbolic group | No | |

solvable group | Yes | The normal closure of is isomorphic to the group , i.e., the group of 2-adic rationals. This is an abelian normal subgroup. The quotient group is isomorphic to , which is an abelian group isomorphic to the group of integers. |

Noetherian group | No | The normal closure of is isomorphic to the subgroup , i.e., the group of 2-adic rationals. This is not finitely generated, because any finite generating set has an upper bound on possible denominators. |

nilpotent group | No | |

abelian group | No | |

polycyclic group | No | |

supersolvable group | No |