show that $D_{2n}=\left \langle a,b \mid a^2=e,b^n=e,aba^{-1}=b^{-1} \right \rangle$ is nilpotent iff $n$ is power of 2.

Question

show that $D_{2n}=\left \langle a,b \mid a^2=e,b^n=e,aba^{-1}=b^{-1} \right \rangle$ is nilpotent iff $n$ is power of $2$.

please help me how should start and proceed,any guide or hint will be great,thanks.

Answer 1

Hint

If $n$ is not a power of $2$, there is an odd prime $p$ dividing $n$. Consider $c = b^{n/p}$, an element of order $p$. Prove that H = $\langle c , a\rangle \le D_{2n}$ is not nilpotent, by showing that $[\langle c \rangle, \langle a \rangle] = \langle c \rangle$.