Subgroup structure of dihedral groups
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.
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:
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:
- Abelian subgroup is isomorph-free in generalized dihedral group unless it is an elementary abelian 2-group
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.
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
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.