Some Subgroup of Dihedral Group is Normal I ran into this question when I was studying for my abstract algebra midterm.

Show that the subgroup $H$ of rotations is normal in the dihedral group $D_n$. Find the quotient group $D_n/H$.

I'm not quite sure where to begin. I know that for a Dihedral group of $n\geq 3$, then $r^n=1$ where $r$ is a rotation, and $s^2=1$ where $s$ is a reflection, and $srs=r^{-1}$. I was not sure how to prove something is a normal subgroup from here. Any advice, thanks!
 A: The index $2$ suggestion works, but you can also show this directly.  One can check that the generators $R$ and $F$ of the dihedral group conform to the rule $RF = FR^{-1}$.  From this, we see that any element in $D_n$ can be written as $R^jF^k$ where $0 \leq j \leq n-1$ and $0 \leq k \leq 1$.  
A subgroup $N \leq G$ is normal whenever, given any $n \in N$ and $g \in G$, we have $gng^{-1} \in N$.  In this case, any element of the rotation subgroup looks like $R^m$ for $1 \leq m \leq n-1$.  Considering any arbitrary element $R^jF^k$ of $D_n$, we just need to show that $(R^jF^k)R^m(R^jF^k)^{-1} \in \langle R \rangle$.  Clearly this is true if $k=0$, so assume $k=1$.  Now look to the helpful rule in the first paragraph to conclude that this is indeed an element of $\langle R \rangle$.
A: If $D_n=\langle r,s\mid r^n=s^2=1,srs=r^{-1}\rangle$, then $D_n$ has order $2n$ and the group generated by $r$ has order $n$.
Therefore the index of $\langle r\rangle$ in $D_n$ is equal to two, and it is a general fact that if $H\leq G$ is a subgroup with $[G:H]=2$ then $H$ is a normal subgroup of $G$.
A: How many rotations are there, and how does this compare to the total number of elements? You may have shown as an exercise previously that a subgroup of index $2$ is normal. That is relevant here. If not, you should prove it, because using this fact is the easiest way I see to solve your problem.
A: This may not count as a proof (depending on your definition of the dihedral group) but it explains geometrically what's going on. The dihedral group is the group of symmetries of the regular $n$-gon in the plane. It consists of $n$ rotations, which clearly form a subgroup, and $n$ reflections. If you think geometrically, the product of a rotation $r$ and a reflection $s$ reverses orientation in the plane, so must be a reflection. 
Note that $grg^{-1}$ is clearly a rotation if $g$ is. It is a product of two reflections if $g$ is a reflection, so preserves orientation and must be a rotation. That says the subgroup of rotations is invariant under conjugation, so normal.
