# D8 in SD16

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) semidihedral group:SD16 (see subgroup structure of semidihedral group:SD16).

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 semidihedral group:SD16, the semidihedral 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 dihedral of order 8 and is given by:

The multiplication table of is as follows:

Element | ||||||||
---|---|---|---|---|---|---|---|---|

## Cosets

The subgroup has index two and is hence a normal subgroup (See index two implies normal). Its left cosets coincide with its right cosets. There are two cosets:

## 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 | |

size of automorphism class of subgroup | 1 |

## Subgroup-defining functions

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

first omega subgroup | subgroup generated by elements of order two | The elements of order two are . These generate this subgroup. |

join of abelian subgroups of maximum rank | subgroup generated by the abelian subgroups that have maximum rank | There are no rank 3 abelian subgroups, and the rank 2 abelian subgroups are and -- both Klein four-groups (see V4 in SD16). Together, they generate the dihedral group. |

join of elementary abelian subgroups of maximum order | subgroup generated by the elementary abelian subgroups of maximum order | There are no elementary abelian subgroups of order 8, and the elementary abelian subgroups of order 4 are and -- both Klein four-groups (see V4 in SD16). Together, they generate the dihedral group. |

## 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 | Yes | On account of being the first omega subgroup, also on account of being an isomorph-free subgroup. |

fully invariant subgroup | invariant under all endomorphisms | Yes | On account of being the first agemo subgroup. |

isomorph-free subgroup | no other isomorphic subgroup | Yes | The other two subgroups of order 8 are Z8 in D16 and Q8 in D16, both non-isomorphic to it. Also follows from being the first omega subgroup. |

homomorph-containing subgroup | contains every homomorphic image | Yes | On account of being an omega subgroup. |

image-closed characteristic subgroup | under any surjective homomorphism from to a group , the image of is characteristic in . | No | If we consider and take the quotient map, then the image of in there looks like one of the Klein four-subgroups of dihedral group:D8, which is not characteristic. |

image-closed fully invariant subgroup | under any surjective homomorphism from to a group , the image of is fully invariant in . | No | Follows from not being image-closed characteristic. |

verbal subgroup | generated by set of words | No | Follows from not being image-closed fully invariant. |

## GAP implementation

The group and subgroup pair can be constructed using GAP as follows:

`G := SmallGroup(16,8); H := Filtered(Subgroups(G), x -> Order(x) = 8 and IsDihedralGroup(x))[1];`

The GAP display is as follows:

gap> G := SmallGroup(16,8); H := Filtered(Subgroups(G), x -> Order(x) = 8 and IsDihedralGroup(x))[1]; <pc group of size 16 with 4 generators> Group([ f4, f3, f2 ])

Here is GAP code to verify some of the assertions on this page:

gap> Order(G); 16 gap> Order(H); 8 gap> Index(G,H); 2 gap> StructureDescription(H); "D8" gap> StructureDescription(G/H); "C2" gap> IsNormal(G,H); true gap> IsCharacteristicSubgroup(G,H); true gap> IsFullinvariant(G,H); true