I'm trying to understand the following part (Chap. Sylow Theorem, Paragraphs preceding the article Application of Sylow Theorem) from Gallian text
I'm trying to understand the why. That is I need to show that $$\forall~g\in A_4,gR_{180}g^{-1}\neq\text{ reflection about any line of symmetry}$$.
Please help me to get that.
You just have to "do" the calculation. Meaning: take any $g$ in the dihedral group of order 12 and do the sequence of symmetries $g^{-1}$, then $R_{180}$, then $g$, and what you'll find is that no matter which $g$ you pick, you'll always find that the resulting composition is always the same as just doing $R_{180}$. Since all rotations commute with each other, it suffices to only consider those $g$ which are reflections, and hopefully you can convince yourself that you need only check one of the reflections about an axis connecting two opposite vertices and one of the reflections about an axis connecting midpoints of two opposite edges.