# Generalized dihedral group for additive group of 2-adic integers

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## Contents

## Definition

### As an abstract group

This group is defined as the generalized dihedral group corresponding to the additive group of 2-adic integers, which is the additive group of -adic integers for the prime .

In other words, it is the external semidirect product of the additive group of 2-adic integers by cyclic group:Z2, where the non-identity element of the acting group acts by the map sending an element to its negative.

### As a profinite group

The group becomes a profinite group under the natural choice of profinite topology. It can also be explicitly realized as the limit of the inverse system:

where each map is (essentially) quotienting out by the center. Explicitly, it can be thought of as reducing the cyclic subgroup of index two modulo a smaller power of 2 while keeping the acting part intact.

## Related groups

- Infinite dihedral group is the generalized dihedral group for the group of integers. This is a dense subgroup of the generalized dihedral group for additive group of 2-adic integers.
- Generalized dihedral group for 2-quasicyclic group has somewhat related and dual properties to this one.

## Group properties

### Abstract group properties

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

p-group | No | The 2-adic integers have infinite order and their order is not a power of 2. |

abelian group | No | semidirect product with nontrivial action |

centerless group | Yes | |

nilpotent group | No | |

hypercentral group | No | |

residually nilpotent group | Yes | |

metabelian group | Yes | |

solvable group | Yes | |

finite group | No | |

finitely generated group | No | |

countable group | No | |

residually finite group | Yes |