# Derived 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:Z4 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 Klein four-group.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 4 and is given by:

The quotient group is a Klein four-group.

## Cosets

The subgroup is a normal subgroup, so its left cosets coincide with its right cosets. There are four cosets:

## Arithmetic functions

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

order of whole group | 16 | |

order of subgroup | 4 | |

index of subgroup | 4 | |

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

number of conjugacy classes in automorphism class | 1 |

## Subgroup-defining functions

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

derived subgroup | subgroup generated by all commutators | The commutators are precisely the elements of this subgroup. For instance, . |

first agemo subgroup | subgroup generated by all powers. Here, , so subgroup generated by squares | |

Frattini subgroup | intersection of all maximal subgroups | |

Jacobson radical | intersection of all maximal normal subgroups |

## Subgroup properties

### Invariance under automorphisms and endomorphisms

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

normal subgroup | invariant under inner automorphisms | Yes | derived subgroup is normal |

characteristic subgroup | invariant under all automorphisms | Yes | derived subgroup is characteristic |

fully invariant subgroup | invariant under all endomorphisms | Yes | derived subgroup is fully invariant, agemo subgroups are fully invariant |

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

verbal subgroup | generated by set of words | Yes | derived subgroup is verbal, agemo subgroups are verbal |

homomorph-containing subgroup | contains every homomorphic image | No | The subgroup is an image of this subgroup but is not contained in it. |

## GAP implementation

The group and subgroup can be constructed using GAP's SmallGroup and DerivedSubgroup functions:

`G := DihedralGroup(16); H := DerivedSubgroup(G);`

The GAP display looks as follows:

gap> G := DihedralGroup(16); H := DerivedSubgroup(G); <pc group of size 16 with 4 generators> Group([ f3, f4 ])Here is a GAP implementation to verify some of the assertions made in this page:[SHOW MORE]