Let $P=M_{p^{n+1}}=\langle a,b \mid a^{p^{n}}=b^p=1, a^b=a^{1+p^{n-1}}\rangle$.
I want to prove that $P^{\prime}=\langle [a,b] \rangle$.
My Try: Clearly $\langle [a,b] \rangle \le P^{\prime}$. Put $K:=\langle [a,b ]\rangle$. Since $a^b=a^{1+p^{n-1}}$, we have $[a,b]=a^{p^{n-1}}$. Thus $K \ {\rm char} \ \langle a \rangle \trianglelefteq P$, and so $K \trianglelefteq P$. Consider $P/K$. If $P/K$ is abelian, then we do. Please help me to complete it.