Suppose $F_1$ and $F_2$ are distinct reflections in $D_n$ such that $F_1$$F_2=F_2F_1$, prove that $F_1F_2=R_{180}$.
I'm stumped on where to even start. Up to the point where I've gotten the book I am using has barely focused on Dihedral groups, yet this is in the chapter questions for basic group properties...
An important relation to keep in mind for dihedral groups is $$ sr^n=r^{-n}s $$ where $r$ and $s$ are elementary rotation and reflection, respectively, and $D_n$ is generated by $r$ and $s$. (You can prove this from the $n=1$ case, $sr=r^{-1}s$.) This can be used to show that the product of reflections is a rotation, since we can write any two reflections as $sr^a$ and $sr^b$ (where $0\leq a,b<n$) and compute: $$ F_1F_2=(sr^a)(sr^b)=(sr^a)(r^{-b}s)=s(r^{a-b})s=ss(r^{b-a})=r^{b-a} $$ using the fact that $s$ has order $2$.
Finally note that $$ (F_1F_2)^2=(F_1F_2)(F_2F_1)=F_1F_2^2F_1=F_1^2=1 $$ since $F_1$ and $F_2$ are reflections and thus have order $2$. We know that $ F_1F_2=F_1F_2^{-1}\neq 1 $ using the fact that each reflection is its own inverse, since $F_1$ and $F_2$ are distinct. This means that $F_1F_2$ is a non-identity rotation whose square is $1$, so $F_1F_2=R_{180}$.