Good evening,
I need some help, with the following task. I know that there are two reflections $t,t'$ with $\langle t, t' \rangle = D_n$.
Now I should tell what I can say generally for two reflections $t,t' \in D_n$ with $t \neq t'$. I know that sometimes the two reflections generate the whole group $D_n$.
But if I look at $D_8=\{id,r,...,r^7,s,sr,...,sr^7\}$ with s=reflection r=rotation then it is not possible to generate every rotation with the two reflections $sr^2$ and $sr^4$. I can't generate $r^7$ for example. I only can generate rotations with an even exponent. If I take $D_3$ then it is possible to generate every element with two reflections, but why?
I read that I can generate dihedral groups $D_m$ with $m|n$. Maybe you can help me to understand that