# Non-normal subgroups of dihedral group:D8

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) cyclic group:Z2 and the group is (up to isomorphism) dihedral group:D8 (see subgroup structure of dihedral group:D8).VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

## Contents

## Definition

Suppose is the dihedral group of order eight (degree four) given by the presentation below, where denotes the identity element of :

.

Then, we are interested in the following four subgroups:

.

and are conjugate subgroups (via , for instance). and are conjugate subgroups (via , for instance). and are not conjugate but are related by an outer automorphism that fixes and sends to . Thus, all four subgroups are automorphic subgroups. These are the only non-normal subgroups of and they are all 2-subnormal subgroups.

## Arithmetic functions

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

order of whole group | 8 | |

order of subgroup | 2 | |

index | 2 | |

size of conjugacy class | 2 | |

number of conjugacy classes in automorphism class | 2 | |

size of automorphism class | 2 | |

subnormal depth | 2 | |

hypernormalized depth | 2 |

## Effect of subgroup operators

Specific values (in the second column) are for .

Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|

normalizer | Klein four-subgroups of dihedral group:D8 | Klein four-group | |

centralizer | Klein four-subgroups of dihedral group:D8 | Klein four-group | |

normal core | -- | trivial group | |

normal closure | Klein four-subgroups of dihedral group:D8 | Klein four-group | |

characteristic core | -- | trivial group | |

characteristic closure | , i.e., | -- | dihedral group:D8 |

## Related subgroups

### Intermediate subgroups

We use here.

Value of intermediate subgroup (descriptive) | Isomorphism class of intermediate subgroup | Small subgroup in intermediate subgroup | Intermediate subgroup in big group |
---|---|---|---|

Klein four-group | Z2 in V4 | Klein four-subgroups of dihedral group:D8 |

## Subgroup properties

### Invariance under automorphisms and endomorphisms

Suppose and denote conjugation by and respectively. Let denote the automorphism that sends to and to . Then, is the inner automorphism group and is the automorphism group.

The automorphism fixes and while interchanging and . The automorphism interchanges and while also interchanging and . The automorphism fixes and while interchanging and . The automorphism interchanges and and also interchanges and .

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

normal subgroup | invariant under inner automorphisms | No | See above description of conjugation automorphisms that permute the subgroups | |

coprime automorphism-invariant subgroup | invariant under automorphisms of coprime order to group | Yes | there are no nontrivial automorphisms of coprime order | |

cofactorial automorphism-invariant subgroup | invariant under all automorphisms whose order has prime factors only among those of the group | No | follows from not being normal | |

2-subnormal subgroup | normal subgroup of normal subgroup | Yes | normal inside Klein four-subgroups of dihedral group:D8 (of the form and ) that are normal in the whole group. | |

subnormal subgroup | Yes | follows from being 2-subnormal, also from being subgroup of nilpotent group. |