# Difference between revisions of "Subgroup structure of dihedral groups"

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+ | {{group family-specific information| | ||

+ | information type = subgroup structure| | ||

+ | group family = dihedral group| | ||

+ | connective = of}} | ||

+ | |||

We study here the subgroup structure of ''finite'' [[dihedral group]]s. See [[subgroup structure of infinite dihedral group]] for the subgroup structure of the [[infinite dihedral group]]. | We study here the subgroup structure of ''finite'' [[dihedral group]]s. See [[subgroup structure of infinite dihedral group]] for the subgroup structure of the [[infinite dihedral group]]. | ||

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There are two kinds of subgroups: | There are two kinds of subgroups: | ||

− | # Subgroups of the form <math>\langle a^d \rangle</math>, where <math>d | n</math>. There is one such subgroup for each <math>d</math>. The total number of such subgroups is <math>\tau(n)</math> or <math>\sigma_0(n)</math>, i.e., the [[divisor count function|number of positive divisors]] of <math>n</math>. | + | # Subgroups of the form <math>\langle a^d \rangle</math>, where <math>d | n</math>. There is one such subgroup for each <math>d</math>. The total number of such subgroups is <math>\tau(n)</math> or <math>\sigma_0(n)</math>, i.e., the [[number:divisor count function|number of positive divisors]] of <math>n</math>. |

− | # Subgroups of the form <math>\langle a^d, a^rx\rangle</math> where <math>d | n</math> and <math>0 \le r < d</math>. There are thus <math>d</math> such subgroups for each such divisor <math>d</math>. The total number of such subgroups is <math>\sigma(n)</math> or <math>\sigma_1(n)</math>, i.e., the [[divisor sum function|sum of positive divisors]] of <math>n</math>. | + | # Subgroups of the form <math>\langle a^d, a^rx\rangle</math> where <math>d | n</math> and <math>0 \le r < d</math>. There are thus <math>d</math> such subgroups for each such divisor <math>d</math>. The total number of such subgroups is <math>\sigma(n)</math> or <math>\sigma_1(n)</math>, i.e., the [[number:divisor sum function|sum of positive divisors]] of <math>n</math>. |

We consider various cases when discussing subgroup structure: | We consider various cases when discussing subgroup structure: | ||

− | * The special cases <math>n = 2, n = 4</math>. | + | * The special cases <math>n = 1, n = 2, n = 4</math>. |

* The special case where <math>n</math> is a power of <math>2</math>. | * The special case where <math>n</math> is a power of <math>2</math>. | ||

* The special case where <math>n</math> is odd. | * The special case where <math>n</math> is odd. | ||

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{{further|[[Cyclic subgroup of dihedral group]]}} | {{further|[[Cyclic subgroup of dihedral group]]}} | ||

+ | |||

+ | ===The special case <math>n = 1</math>=== | ||

+ | |||

+ | In the case <math>n = 1</math>, the subgroup <math>\langle a \rangle</math> is trivial, and the whole group is [[cyclic group:Z2|cyclic of order two]] generated by <math>x</math>. | ||

===The special case of <math>n = 2</math>=== | ===The special case of <math>n = 2</math>=== | ||

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* [[Prehomomorph-contained subgroup]]: {{proofat|[[Cyclic subgroup is prehomomorph-contained in dihedral group]]}} | * [[Prehomomorph-contained subgroup]]: {{proofat|[[Cyclic subgroup is prehomomorph-contained in dihedral group]]}} | ||

− | * [[Isomorph-free subgroup]]: {{proofat|[[Cyclic subgroup is isomorph- | + | * [[Isomorph-containing subgroup]] (for finite <math>n</math>, this is equivalent to [[isomorph-free subgroup]]): {{proofat|[[Cyclic subgroup is isomorph-containing in dihedral group]]}} |

* [[Characteristic subgroup]]: {{proofat|[[Cyclic subgroup is characteristic in dihedral group]]}} | * [[Characteristic subgroup]]: {{proofat|[[Cyclic subgroup is characteristic in dihedral group]]}} | ||

− | ===The case <math>n \ne 4</math>=== | + | These facts have a number of generalizations: |

+ | |||

+ | * [[Abelian subgroup is isomorph-free in generalized dihedral group unless it is an elementary abelian 2-group]] | ||

+ | |||

+ | ===The case <math>n \ne 1,2, 4</math>=== | ||

+ | |||

+ | In the case <math>n \ne 2, 4</math>, the subgroup <math>\langle a \rangle</math> is the [[centralizer of commutator subgroup]], i.e., it is the centralizer in <math>D_{2n}</math> of the [[commutator subgroup]] of <math>D_{2n}</math>, which is <math>\langle a^2 \rangle</math>. | ||

+ | |||

+ | There are a number of generalizations/related facts: | ||

− | In the | + | * [[Commutator subgroup centralizes cyclic normal subgroup]]: In particular, the cyclic part in a dihedral group is contained in the [[centralizer of commutator subgroup]] for all <math>n</math>. The cases <math>n = 2,4</math> need to be excluded because these are the only cases where the centralizer of commutator subgroup is ''bigger'', i.e., the whole group. |

+ | * [[Abelian subgroup equals centralizer of commutator subgroup in generalized dihedral group unless it is a 2-group of exponent at most four]] | ||

+ | * [[Abelian subgroup is contained in centralizer of commutator subgroup in generalized dihedral group]] | ||

===Odd versus even <math>n</math>=== | ===Odd versus even <math>n</math>=== |

## Latest revision as of 20:14, 9 September 2009

This article gives specific information, namely, subgroup structure, about a family of groups, namely: dihedral group.

View subgroup structure of group families | View other specific information about dihedral group

We study here the subgroup structure of *finite* dihedral groups. See subgroup structure of infinite dihedral group for the subgroup structure of the infinite dihedral group.

For any natural number , we define:

.

For , is the group of symmetries in the regular -gon in the plane.

There are two kinds of subgroups:

- Subgroups of the form , where . There is one such subgroup for each . The total number of such subgroups is or , i.e., the number of positive divisors of .
- Subgroups of the form where and . There are thus such subgroups for each such divisor . The total number of such subgroups is or , i.e., the sum of positive divisors of .

We consider various cases when discussing subgroup structure:

- The special cases .
- The special case where is a power of .
- The special case where is odd.

## The subgroup

`Further information: Cyclic subgroup of dihedral group`

### The special case

In the case , the subgroup is trivial, and the whole group is cyclic of order two generated by .

### The special case of

In the case , the group is the Klein four-group:

.

For this, the subgroup is a normal subgroup, but not a characteristic subgroup.

### The case

In the case , the subgroup is the unique cyclic subgroup of order . It satisfies the following properties:

- Prehomomorph-contained subgroup:
`For full proof, refer: Cyclic subgroup is prehomomorph-contained in dihedral group` - Isomorph-containing subgroup (for finite , this is equivalent to isomorph-free subgroup):
`For full proof, refer: Cyclic subgroup is isomorph-containing in dihedral group` - Characteristic subgroup:
`For full proof, refer: Cyclic subgroup is characteristic in dihedral group`

These facts have a number of generalizations:

### The case

In the case , the subgroup is the centralizer of commutator subgroup, i.e., it is the centralizer in of the commutator subgroup of , which is .

There are a number of generalizations/related facts:

- Commutator subgroup centralizes cyclic normal subgroup: In particular, the cyclic part in a dihedral group is contained in the centralizer of commutator subgroup for all . The cases need to be excluded because these are the only cases where the centralizer of commutator subgroup is
*bigger*, i.e., the whole group. - Abelian subgroup equals centralizer of commutator subgroup in generalized dihedral group unless it is a 2-group of exponent at most four
- Abelian subgroup is contained in centralizer of commutator subgroup in generalized dihedral group

### Odd versus even

When is odd, the cyclic subgroup of order in the dihedral group of order , satisfies the following properties:

- Normal Hall subgroup
- Verbal subgroup
- Fully invariant subgroup
- Order-containing subgroup
- Homomorph-containing subgroup
- Image-closed fully invariant subgroup
- Image-closed characteristic subgroup

None of these properties are satisfied when is even.

Also, when is odd, the cyclic subgroup of order is the commutator subgroup.

## The subgroup

### General facts

Here are some general facts about this subgroup.

- Its normalizer is when is even. When is odd, is self-normalizing.
- The conjugate subgroups to this subgroup are subgroups of the form . When is odd, this includes
*all*subgroups of the form , whereas if is even, this includes only*half*the subgroups of the form . There are thus conjugate subgroups if is even and conjugate subgroups if is odd. - If ,there are three automorphic subgroups, . If , the automorphic subgroups to this subgroup are subgroups of the form . There are thus of them if .
- The normal closure of is . This is the whole group if is odd and is a subgroup of index two if is even.

### The special case of

When , the dihedral group is the Klein four-group, and is a normal subgroup. There is *no other* for which is a normal subgroup.

### The case

Some properties satisfied whenever are:

### Odd versus even

When is odd, then the subgroup satisfies the following properties:

- Sylow subgroup, Hall subgroup
- Order-conjugate subgroup, isomorph-conjugate subgroup, automorph-conjugate subgroup
- Contranormal subgroup
- Self-normalizing subgroup

On the other hand, when is even, this subgroup satisfies none of these properties.

### The case where is a power of two

When is a power of two, the subgroup satisfies the following properties:

- Subnormal subgroup: In particular, when , is a -subnormal subgroup.