# Changes

## Subgroup structure of dihedral groups

, 20:14, 9 September 2009
no edit summary
{{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 $\langle a^d \rangle$, where $d | n$. There is one such subgroup for each $d$. The total number of such subgroups is $\tau(n)$ or $\sigma_0(n)$, i.e., the [[number:divisor count function|number of positive divisors]] of $n$.# Subgroups of the form $\langle a^d, a^rx\rangle$ where $d | n$ and $0 \le r < d$. There are thus $d$ such subgroups for each such divisor $d$. The total number of such subgroups is $\sigma(n)$ or $\sigma_1(n)$, i.e., the [[number:divisor sum function|sum of positive divisors]] of $n$.
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 $n \ne 1,2, 4$=== 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 case [[centralizer of commutator subgroup]] for all $n \ne 4$, the subgroup . The cases $\langle a \ranglen = 2,4$ 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 $D_{2n}$ 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 $D_{2n}$, which is $\langle a^2 \rangle$.
===Odd versus even $n$===