# Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$}

Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a subgroup of $\langle R \rangle$ or $n$ is even and $N$ is one of $\langle r,R^2 \rangle$ or $\langle rR,R^2 \rangle$ .

How can I prove this theorem?

First prove $\langle R^k\rangle \unlhd D_n\ \forall\ k\in\mathbb{Z}$, which is relatively easy and solves the case $N\leq \langle R\rangle$.
Second, prove that $N \unlhd D_n$ and $rR^j \in N$ for some $j$ implies $r^2 \in N$, which solves the case "$n$ is even and..."
In this answer to your previous question I explained how to manipulate conjugacy equations in dihedral groups. Figure out what the proper nontrivial subgroups of $D_n$ are (hint: this depends on when $n$ is even or odd), then apply the method I used in that answer to figure out which are invariant under conjugation.
• I know that when $n$ is odd, $⟨r,R^2⟩ and ⟨rR,R^2⟩$ will generate the whole group, so it can't be the proper subgroup, can it? </br> when $n$ is even $⟨r,R^2⟩ and ⟨rR,R^2⟩$ will generate part of the group $D_n$, so it's a proper subgroup of $D_n$. But how does this relate to this theorem. – Diane Vanderwaif Apr 12 '13 at 12:12