# Klein four-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) Klein four-group and the group is (up to isomorphism) dihedral group:D8 (see subgroup structure of dihedral group:D8).

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

This article discusses the dihedral group of order eight (see details on the subgroup structure) and the two Klein four-subgroups of this group. We call the dihedral group , and use the following presentation:

and we set:

.

As full lists:

## Contents

## Cosets

Both subgroups have index two and are hence normal subgroups. Each of them has two cosets: the subgroup itself and the rest of the group.

The cosets of are:

The cosets of are:

## 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)

has two permutable complements: and , both of which are conjugate subgroups and subgroups of . has two permutable complements: and , both of which are conjugate subgroups and are subgroups of .

The four subgroups of order two arising as complements in this way are all automorphic subgroups. For information on these, see non-normal subgroups of dihedral group:D8.

For convenience, we refer to below, though behaves the same way.

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 | 8 | |

order of subgroup | 4 | |

index | 2 | |

size of conjugacy class | 1 | |

number of conjugacy classes in automorphism class | 2 |

## Effect of subgroup operators

In the table below, we provide values specific to .

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

normalizer | whole group | dihedral group:D8 | |

centralizer | the subgroup itself | current page | Klein four-group |

normal core | the subgroup itself | current page | Klein four-group |

normal closure | the subgroup itself | current page | Klein four-group |

characteristic core | center of dihedral group:D8 | cyclic group:Z2 | |

characteristic closure | whole group | dihedral group:D8 | |

commutator with whole group | center of dihedral group:D8 | cyclic group:Z2 |

## Automorphism class-defining functions

Below are some functions that start with the whole group as a black box group and give the unique *automorphism class* comprising the two Klein four-subgroups. Note that these functions are not guaranteed to always give a single automorphism class of subgroups for a general group.

Function | Where does it make sense in general? | Always gives unique automorphism class? | Why it gives these subgroups? |
---|---|---|---|

elementary abelian subgroup of maximum order | typically, context of group of prime power order | No | By inspection of subgroup structure |

abelian subgroup of maximum rank | typically, context of group of prime power order | No | By inspection of subgroup structure |

non-characteristic normal subgroup | any group | No | By inspection of subgroup structure |

## Related subgroups

### Intermediate subgroups

There are no properly intermediate subgroups because the subgroup is a maximal subgroup.

### Smaller subgroups

Below we discuss subgroups inside .

Value of smaller subgroup (descriptive) | Isomorphism class of smaller subgroup | Smaller subgroup in subgroup | Smaller subgroup in whole group |
---|---|---|---|

cyclic group:Z2 | Z2 in V4 | center of dihedral group:D8 | |

cyclic group:Z2 | Z2 in V4 | non-normal subgroups of dihedral group:D8 | |

cyclic group:Z2 | Z2 in V4 | non-normal subgroups of dihedral group:D8 |

### Images under quotient maps

The discussion below is for .

Kernel of quotient map (descriptive) | Kernel of quotient map (link) | Image of whole group as group | Image of subgroup as group | Embedding (link) |
---|---|---|---|---|

cyclic maximal subgroup of dihedral group:D8 | cyclic group:Z2 | cyclic group:Z2 | Z2 in itself | |

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

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

## 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. It turns out that and , while and preserve both and . Note that the automorphism group is isomorphic to dihedral group:D8 and the inner automorphism group to the Klein four-group.

Note that since sends to , they both satisfy the same subgroup properties. For convenience, we use notation and symbols for .

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

normal subgroup | invariant under inner automorphisms | Yes | subgroup of index two is normal | |

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

coprime automorphism-invariant normal subgroup | normal and coprime automorphism-invariant | Yes | ||

characteristic subgroup | invariant under all automorphisms | No | ||

cofactorial automorphism-invariant subgroup | invariant under all automorphisms whose order has prime factors only among those of the group | No | has order two, and | |

fully invariant subgroup | invariant under all endomorphisms | No | (via characteristic) |

### Advanced properties based on invariance/resemblance

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

order-normal subgroup | every subgroup of the same order is normal | Yes | maximal subgroup of group of prime power order is normal | |

isomorph-normal subgroup | every isomorphic subgroup is normal | Yes | ||

isomorph-automorphic subgroup | every isomorphic subgroup is automorphic | Yes | ||

order-isomorphic subgroup | isomorphic to all subgroups of the same order | No | Cyclic subgroup has same order, not isomorphic |

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

central factor | product with centralizer is whole group | No | The subgroup is a proper self-centralizing subgroup. |

conjugacy-closed subgroup | any two elements of the subgroup conjugate in the whole group are conjugate in the subgroup | No | The elements and are conjugate in the whole group and not in the subgroup, which is abelian. |

transitively normal subgroup | every normal subgroup of the subgroup is normal in the whole group | No | The subgroup (see non-normal subgroups of dihedral group:D8) inside is normal in , not in . See also normality is not transitive -- this is the smallest example. |

## Generic maximality notions

and are both maximal subgroups of the group of prime power order . Thus, they satisfy all these properties: maximal normal subgroup, maximal subgroup, subgroup of index two, order-normal subgroup, isomorph-normal subgroup, maximal subgroup of finite nilpotent group.

As already discussed, they are both also coprime automorphism-invariant, hence they are isomorph-normal coprime automorphism-invariant subgroup of group of prime power order. In particular, they are both fusion system-relatively weakly closed subgroups and thus Sylow-relatively weakly closed subgroups.

### Abelian subgroups of maximum order

and are both abelian subgroups of maximum order. There is one more abelian subgroup of maximum order, namely the cyclic subgroup .

### Elementary abelian subgroups of maximum order

and are both elementary abelian subgroups of maximum order in . There are no others.

### Abelian subgroups of maximum rank

and are both abelian subgroups of maximum rank in . There are no others.

## GAP implementation

### Finding these subgroups inside a black-box dihedral group

Suppose is a group that we know is isomorphic to dihedral group:D8. We can create a two-element list of the Klein four-subgroups of using the command:

H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [4,2]);

We can then set:

H1 := H[1]; H2 := H[2];

### Constructing the dihedral group and the two subgroups

This can be achieved by the sequence of commands:

G := DihedralGroup(8); H := Filtered(NormalSubgroups(G), x -> IdGroup(x) = [4,2]); H1 := H[1]; H2 := H[2];