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.