2D figure with a nontrivial rotational symmetry

Question

I am stuck with following problem, could anyone help me?

(1) Can a finite 2D figure with a nontrivial rotational symmetry can have exactly one reflection symmetry?

thanks

Answer 1

No. Assume the figure has one reflection symmetry about a vertical axis. Since the figure is finite, let the topmost point of the figure's boundary intersect this axis at point A. Suppose a forward rotation carries point A to point B. CONJUGATE: Perform a backward rotation carrying B to A, apply the reflective symmetry (which leaves A fixed!), and perform a forward rotation carrying A back to B. This is a reflective symmetry with axis passing through B, which is nearly always a different line. If B is also on the vertical axis, the rotation is a 180-degree rotation, which can always also be achieved by two perpendicular reflections, e.g about the vertical axis and about a horizontal axis.

Answer 2

Let $R_1$ be a nontrivial rotation that preserves the figure. Let $\phi_1$ be a reflection that preserves the figure.

What is $(\phi_1 R_1)^{-1}$?