In Dihedral group $D_n$ write the following in the form $r^i$ or $r^i f$ where $0\leq i < n$

Question

In Dihedral group $D_n$ write the following in the form $r^i$ or $r^i f$ where $0\leq i < n$. (r is rotation and f is reflection, I think)

Also given, $r=R_{360/n}$

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

b) In $D_5$, $r^{-3}fr^4fr^{-2}$

c) In $D_6$, $fr^{5}fr^{-2}f$

for (a)

We know in $D_4$, if $r=R_{360/n}$ then $r^n = e$ and also $f^2=e$

$fr^{-2}fr^5$ = $fr^2fr$

now $r^2$ = $R_{180}$ in $D_4$ which commutes, hence

=$r^2f^2 r$ = $r^3$

I was able to solve (a) because I am familiar with $D_4$ but am getting stuck in (b) and (c)

Answer 1

For any dihedral group $frf = r^{-1}$ which implies $fr^i = r^{-i}f$ for all $i$.

Thus a more mechanical solution for a) which doesn't depend on $n$ until you get to the end and thus works for the other parts would be $fr^{-2}fr^5=r^2f^2r^5=r^2r^5=r^7=r^3$