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.


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 $\langle R^2,RF \rangle$.

  • 1
    $\begingroup$ You have lost track of the group $D_n$ itself. $\endgroup$ – Alex M. Jun 12 '16 at 10:37
  • $\begingroup$ Yes, you are right. Of course, the group itself should be included. $\endgroup$ – Dietrich Burde Jun 12 '16 at 11:39
  • 2
    $\begingroup$ By saying $\langle{R^d}\rangle$ is a normal subgroup for all divisors $d\ \mid\ {n}$, you've actually already included {1}, because $n\ \mid\ {n}$. $\endgroup$ – Rasputin Oct 31 '16 at 20:22
  • $\begingroup$ @ Dietrich Burde : Is there any other normal subgroups of $D_{2n}$ besides these normal subgroups $\endgroup$ – user120386 May 22 '17 at 10:08
  • 1
    $\begingroup$ @user120386 No, this is a complete classification of normal subgroups, see Keith Conrad's article on dihedral groups. $\endgroup$ – Dietrich Burde May 22 '17 at 13:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.