I solved the following exercise:

In the dihedral group $D_n$ let $R = R_{360/n}$ and let $F$ be any reflection. Write each of the following products in the form $R^i$ or $R^iF$:

(a) In $D_4$, $FR^{-2}FR^5$

I solved this exercise by noting that $R^{-2}$ is $R^2$ and then I picked a fixed $F$ and calculated the product. The answer is $R^3$.

However this is unsatisfying as I'd much prefer to solve this analytically, that is, without calculating an example.

So my question is:

What's the clever way to solve this exercise? What observations about such products can I make to simplify them without example calculations?

• Because your question is about $D_4$ you can also observe that $R^2$ is rotation by 180 degrees, i.e. multiplication of vectors by $-1$. It follows that $R^2$ commutes with all linear transformations. In particular $R^{-2}F=FR^{-2}$. You will also need $F^2=1$. If you need this for $n\neq4$, then follow Matt Samuel's recipe (+1). – Jyrki Lahtonen Oct 29 '15 at 7:25

$F^2=1$, $R^n=1$, $FRF=R^{-1}$ is all you need. For example $$(FR^{-2}F)R^5=R^2R^5=R^7$$ Which verifies your calculation when the order is 4.
• Thank you, perfect answer, $FRF=R^{-1}$ was what I was missing! – a student Nov 3 '15 at 0:43