Let $p> 3$ be a prime number and $G$ be a finite group of order $p(p^2-1) / 4 $. If $ 4 \mid (p+1)$ then easily we can see that the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. As I checked it seems that for each $p$ this result holds i.e. the $ p$-Sylow subgroup of $G$ is a normal subgroup of $G$. Any references or hints would be highly appreciated. Thanks in advance.



groups of order $ p(p^2-1) / 4 $ where $p$ is a prime
If the Sylow $p$-subgroup was not normal, then it is not hard to see that the only possibility would be $4|(p-1)$ with $p+1$ Sylow $p$-subgroups. Then $G$ would act $2$-transitively by conjugation on the $p+1$ Sylow $p$-subgroups, with $2$-point stabilizer a factor of $(p-1)/4$.


The finite $2$-transitive permutations are all known using the classification of finite simple groups, and there are complete lists available, for example in Dixon and Mortimer's book on permutation groups. There are none with these properties. Note that the affine case cannot occur, because $p+1$ cannot be a prime power.


I don't know whether this could be proved without using CFSG.


