I have some difficulties finding the elements of Dihedral Group $D_8$ (note that the order $|D_8|=8$). I know the geometric approach for defining $D_8$, but I prefer the algebraic way. In this case, I'd like to work with the group presentation.
Namely, $D_8=\langle x,a|a^n=x^2=e, xax^{-1}=a^{-1} \rangle$ with $n=4$. My question is about what the elements are in $D_8$. I really want to know why $a^3$ is not equal to $1$. I tried this work with this way: firstly, I consider the Dihedral Group $D_8 = X / S$, where $X$ is the free group generated by $\{x,a\}$, and $S$ is the smallest normal subgroup of $\langle x,a\rangle$ that contains $\{a^4,x^2,axax^{-1}\}$. Thus I want to figure out whether $a^3$ is in $S$ or not.
I agree $a^3$ is most likely not in $S$, but I want to make sure that it is true in very rigorous way. And I want to generalize this way for the general case, $D_{2n}$.
If I violated some rules of this awesome forum, Please let me know so that I can fix it.
Thanks.