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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.

Community
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Pteromys
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1 Answers1

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

Dietrich Burde
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    You have lost track of the group $D_n$ itself. – Alex M. Jun 12 '16 at 10:37
  • Yes, you are right. Of course, the group itself should be included. – Dietrich Burde Jun 12 '16 at 11:39
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    By saying $\langle{R^d}\rangle$ is a normal subgroup for all divisors $d\ \mid\ {n}$, you've actually already included {1}, because $n\ \mid\ {n}$. – Rasputin Oct 31 '16 at 20:22
  • @ Dietrich Burde : Is there any other normal subgroups of $D_{2n}$ besides these normal subgroups – user120386 May 22 '17 at 10:08
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    @user120386 No, this is a complete classification of normal subgroups, see Keith Conrad's article on dihedral groups. – Dietrich Burde May 22 '17 at 13:51
  • I'd like to prove that argument for some given $n$ and I understand that this answer provides a way to find normal subgroups of $D_n$ but how do you prove that this are the only subgroups of $D_n$? – Noa Even Jan 11 '21 at 12:56
  • @Jneven those are not the only subgroups of $D_n$: they are the only normal subgroups of $D_n$. There are non-normal subgroups of $D_n$ as well, which are not included in the answer above, such as $\langle F\rangle = {1,F}$. At the link provided in the answer, Theorem 3.1 describes all subgroups of $D_n$ while Theorem 3.8 describes all normal subgroups of $D_n$. – KCd Jun 18 '22 at 13:45