Let $G$ be a group of order $p(p^2+1)$, where $p$ is an odd prime number and $p>3$. Easily we can see that $G$ is solvable and so $G$ has a Hall subgroup $L$ of order $p^2+1$. Also we know that $P$, the Sylow subgroup of $G$ is a normal subgroup of $G$. Is it true in general that $L$ is a normal subgroup of $G$? 



On the groups of order $p(p^2+1)$
As Jeremy Rickard points out, the answer is no in general. However, $G$ has order divisible by $2,$ but not by $4.$ It follows that $G$ has a normal $2$-complement $K$ of order $p \left( \frac{p^{2}+1}{2} \right).$ Now $K$ has a normal $p$-complement by Burnside's transfer theorem, since ${\rm gcd}(p-1,\frac{p^{2}+1}{2}) = 1.$ This subgroup of $K$ has order $\frac{p^{2}+1}{2}$ and is characteristic in $K,$ hence normal in $G.$ Thus there is a subgroup of index $2$ in $L$ which is normal in $G.$


