I'm trying to solve this problem from my group theory course:
Consider the dihedral group $D_6$ of isometries of the euclidean plane which fix a regular hexagon:
(a) Prove that this group is generated by the clockwise rotation $x$ of angle $\frac{\pi}{3}$ and by the reflection $y$ through two opposite vertices.
(b) Show that $x^6=1=y^2$ and $yx=x^{-1}y$.
(c) Show that $D_6=\{x^ny^m:0\leq n<6, 0\leq m<2\}$
I'm having trouble finding good answers for (a) and (c) (the question (b) is very easy to prove). I have the idea in my head, I know that's true, my problem is about finding a "right" way to explain it on paper. I don't know what are the right arguments to prove that statements are true. In fact, I have this problem in general, every time I need to prove some elements generate a certain group. How can I prove this? How can I prove that certain elements generate certain group in general? Any help will be appreciated, thanks in advance.