Extra-Special $p$ group and complement Let $G$ be an extra special $p$ group of order $p^{2n+1}$, $n\geq 2$. Does $[G,G]$ necessarily have a complement in $G$? I dont think so, but I am not sure.
Sorry this should be a very silly question, but I do not have much knowledge in $p$ group theory.

$[G,G]$ never has a complement.  For in an extraspecial group, $[G,G] \cong C_p$, a cyclic group of order $p$.  If $[G,G]$ has a complement $H$, then by the definition of an extraspecial group, $H \cong C_p^{2n}$.  Furthermore, $H$ must act trivially on $[G,G]$, because $H$ has order a power of $p$, while the automorphism group of $C_p$ has order $p-1$.  Then $G \cong [G,G] \times H$, so $G$ is abelian, a contradiction.
