Compute the order of each of the elements in $D_6$ where $D_{6}=\left\langle r, s \mid r^{3}=s^{2}=1, r s=s r^{-1}\right\rangle$

Question

Compute the order of each of the elements in $D_6$ where $D_{6}=\left\langle r, s \mid r^{3}=s^{2}=1, r s=s r^{-1}\right\rangle$

I found six elements of $D_6$ are $1,r, r^2,s, rs, r^2s.$

How can I show that there is no element other than $1,r, r^2,s, rs, r^2s$?

Answer 1

For the order of $rs$ we have \begin{eqnarray*} (rs)^2&=&rsrs\\ &=&sr^{-1}rs\\ &=&s^2\\ &=&1 \end{eqnarray*}

Similarly for $r^2s$.