Are groups of order $p(p + 1)$ solvable nilpotent?
I proved before that groups of order $p(p + 1)$ have a normal subgroup of order $p$ or a normal subgroup of order $p + 1$.
In that case if we have normal subgroup $H$ of order $p + 1$, then $G$ is nilpotent since the series $1 \lhd H \lhd G$ and the factors are abelian. However, what about solvability and nilpotent in the other cases ?