# D8 in D16

From Groupprops

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) dihedral group:D8 and the group is (up to isomorphism) dihedral group:D16 (see subgroup structure of dihedral group:D16).

The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on particular subgroups in groups

## Contents

## Definition

Here, is the dihedral group:D16, the dihedral group of order sixteen (and hence, degree eight). We use here the presentation:

has 16 elements:

The subgroups and of interest are:

Both subgroups are normal but are related by the outer automorphism class of .

## Cosets

Both subgroups has index two and are normal subgroup (See index two implies normal), so left cosets coincide with right cosets.

The cosets of are:

The cosets of are:

## Arithmetic functions

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

order of whole group | 16 | |

order of subgroup | 8 | |

index of subgroup | 2 | |

size of conjugacy class (=index of normalizer) | 1 | |

number of conjugacy classes in automorphism class | 2 |

## Subgroup properties

### Invariance under automorphisms and endomorphisms

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

normal subgroup | invariant under inner automorphisms | Yes | index two implies normal |

characteristic subgroup | invariant under all automorphisms | No | The automorphism interchanges and . |

fully invariant subgroup | invariant under all endomorphisms | No | (follows from not being characteristic) |

isomorph-free subgroup | no other isomorphic subgroup | No | The two subgroups are isomorphic. |

isomorph-automorphic subgroup | all isomorphic subgroups are automorphic subgroups | Yes | are the only subgroups isomorphic to dihedral group:D8. |

potentially fully invariant subgroup | fully invariant subgroup inside a possibly bigger group | No | See D8 is not potentially fully invariant in D16. |