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Subgroup structure of dihedral groups

534 bytes added, 13:29, 2 September 2009
The case n \ne 4
* [[Characteristic subgroup]]: {{proofat|[[Cyclic subgroup is characteristic in dihedral 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 [[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 unless it is a 2-group of rank at most two]]
===Odd versus even <math>n</math>===
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