Dihedral Group $D_{2016}$. Simplify $a^3ba^2baba^{-1}$ in terms of $b^ra^m$, where r and m are non negative integers.

Question

We know that $a^{1008}=b^2=e$ and $a^jb=ba^{-j}$ for all $j$

I keep getting the answer $ba^{-3}$ but $-3<0$ so it is negative.

How should I do this properly?

Answer 1

From $a^{1008}=e$, you can get that $a^{1005}=a^{-3}$.
Hence $ba^{-3}=ba^{1005}$ which fulfill the requirement of your problem.