# Changes

## Subgroup structure of dihedral groups

, 13:33, 2 September 2009
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We consider various cases when discussing subgroup structure:
* The special cases $n = 1, n = 2, n = 4$.
* The special case where $n$ is a power of $2$.
* The special case where $n$ is odd.
{{further|[[Cyclic subgroup of dihedral group]]}}

===The special case $n = 1$===

In the case $n = 1$, the subgroup $\langle a \rangle$ is trivial, and the whole group is [[cyclic group:Z2|cyclic of order two]] generated by $x$.
===The special case of $n = 2$===
* [[Prehomomorph-contained subgroup]]: {{proofat|[[Cyclic subgroup is prehomomorph-contained in dihedral group]]}}
* [[Isomorph-containing subgroup]] (for finite $n$, 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 $n \ne 1,2, 4$===
* [[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 $n$. The cases $n = 2,4$ 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 $n$===