# Normal subgroups of dihedral groups

In relation to my previous question, I am curious about what exactly are the normal subgroups of a dihedral group $D_n$ of order $2n$.

It is easy to see that cyclic subgroups of $D_n$ is normal. But I suspect that case analysis is needed to decide whether dihedral subgroups of $D_n$ is normal.

A little bit of Internet search suggests the use of semidirect product $(\mathbb Z/n\mathbb Z) \rtimes (\mathbb Z/2\mathbb Z) \cong D_n$, but I do not know the condition for subgroups of a semidirect product to be normal.

I would be grateful if you could suggest a way to enumerate the normal subgroups of $D_n$ that does not resort to too much of case analysis.

## 1 Answer

Here is a nice answer: the dihedral group is generated by a rotation $$R$$ and a reflection $$F$$ subject to the relations $$R^n=F^2=1$$ and $$(RF)^2=1$$. For $$n$$ odd the normal subgroups are given by $$D_n$$ and $$\langle R^d \rangle$$ for all divisors $$d\mid n$$. If $$n$$ is even, there are two more normal subgroups, i.e., $$\langle R^2,F \rangle$$ and $$\langle R^2,RF \rangle$$.

• You have lost track of the group $D_n$ itself. Jun 12, 2016 at 10:37
• Yes, you are right. Of course, the group itself should be included. Jun 12, 2016 at 11:39
• By saying $\langle{R^d}\rangle$ is a normal subgroup for all divisors $d\ \mid\ {n}$, you've actually already included {1}, because $n\ \mid\ {n}$. Oct 31, 2016 at 20:22
• @ Dietrich Burde : Is there any other normal subgroups of $D_{2n}$ besides these normal subgroups May 22, 2017 at 10:08
• @user120386 No, this is a complete classification of normal subgroups, see Keith Conrad's article on dihedral groups. May 22, 2017 at 13:51