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Subgroup structure of dihedral groups

512 bytes added, 13:33, 2 September 2009
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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>===
* [[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 rank at most two]]
===Odd versus even <math>n</math>===
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