# Subgroup structure of dihedral groups

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.

## Contents

## 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 rank at most two

### 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.