# Cyclic maximal subgroup of dihedral group: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) cyclic group:Z8 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 subgroup of interest is the subgroup . It is cyclic of order 8 and is given by:

## Cosets

The subgroup has index two and is hence normal (since index two implies normal). Its left cosets coincide with its right cosets, and there are two cosets:

## Complements

COMPLEMENTS TO NORMAL SUBGROUP: TERMS/FACTS TO CHECK AGAINST:TERMS: permutable complements | permutably complemented subgroup | lattice-complemented subgroup | complemented normal subgroup (normal subgroup that has permutable complement, equivalently, that has lattice complement) | retract (subgroup having a normal complement)FACTS: complement to normal subgroup is isomorphic to quotient | complements to abelian normal subgroup are automorphic | complements to normal subgroup need not be automorphic | Schur-Zassenhaus theorem (two parts: normal Hall implies permutably complemented and Hall retract implies order-conjugate)

There are eight possible permutable complements to in , all of them automorphic to each other:

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

complemented normal subgroup | normal subgroup with permutable complement | Yes | see above | |

permutably complemented subgroup | subgroup with permutable complement | Yes | (via complemented normal) | |

lattice-complemented subgroup | subgroup with lattice complement | Yes | (via permutably complemented) | |

retract | has a normal complement | No | ||

direct factor | normal subgroup with normal complement | No |

### 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 | 1 |

## Effect of subgroup operators

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

normalizer | whole group | -- | dihedral group:D16 |

centralizer | -- the subgroup itself | current page | cyclic group:Z8 |

normal core | the subgroup itself | current page | cyclic group:Z8 |

normal closure | the subgroup itself | current page | cyclic group:Z8 |

characteristic core | the subgroup itself | current page | cyclic group:Z8 |

characteristic closure | the subgroup itself | current page | cyclic group:Z8 |

## Subgroup-defining functions

Subgroup-defining function | Meaning in general | Why it takes this value |
---|---|---|

centralizer of derived subgroup | centralizer of the derived subgroup (the commutator of the group with itself) | The derived subgroup is , which is centralized by all of and by nothing outside it. |

## Subgroup properties

### Invariance under automorphisms and endomorphisms

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

normal subgroup | invariant under inner automorphisms | Yes | |

characteristic subgroup | invariant under all automorphisms | Yes | On account of being the centralizer of derived subgroup, also on account of being an isomorph-free subgroup. |

fully invariant subgroup | invariant under all endomorphisms | No | not invariant under the retraction with kernel and image . |

isomorph-free subgroup | no other isomorphic subgroup | Yes |