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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]].
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 [[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 [[number:divisor sum function|sum of positive divisors]] of <math>n</math>.
We consider various cases when discussing subgroup structure:
* 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 odd.
{{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>===
* [[Prehomomorph-contained subgroup]]: {{proofat|[[Cyclic subgroup is prehomomorph-contained in dihedral group]]}}
* [[Isomorph-containing subgroup]] (for finite <math>n</math>, this is equivalent to [[isomorph-free subgroup]]): {{proofat|[[Cyclic subgroup is isomorph-free containing in dihedral group]]}}
* [[Characteristic subgroup]]: {{proofat|[[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]] ===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:
* [[Commutator subgroup centralizes cyclic normal subgroup]]: In particular, the cyclic part in a dihedral group is contained in the case [[centralizer of commutator subgroup]] for all <math>n \ne 4</math>, the subgroup . The cases <math>\langle a \ranglen = 2,4</math> is need to be excluded because these are the only cases where the [[centralizer of commutator subgroup]]is ''bigger'', i.e., it is the whole group.* [[Abelian subgroup equals centralizer of commutator subgroup in <math>D_{2n}</math> generalized dihedral group unless it is a 2-group of the exponent at most four]]* [[Abelian subgroup is contained in centralizer of commutator subgroupin generalized dihedral group]] of <math>D_{2n}</math>, which is <math>\langle a^2 \rangle</math>.
===Odd versus even <math>n</math>===
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