# Changes

## Subgroup structure of dihedral groups

, 13:29, 2 September 2009
The case n \ne 4
* [[Characteristic subgroup]]: {{proofat|[[Cyclic subgroup is characteristic in dihedral group]]}}
===The case $n \ne 1,2, 4$===
In the case $n \ne 2, 4$, the subgroup $\langle a \rangle$ is the [[centralizer of commutator subgroup]], i.e., it is the centralizer in $D_{2n}$ of the [[commutator subgroup]] of $D_{2n}$, which is $\langle a^2 \rangle$. 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 $n$. 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 unless it is a 2-group of rank at most two]]
===Odd versus even $n$===