groups of order $ p(p^2-1) / 4 $ where $p$ is a prime

Question

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.

Answer 1

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.

Answer 2

As a matter of historical interest, R. Brauer classified finite groups of order divisible by a prime $p$ to the first power, but of order less than $p^{3}$ for some prime $p,$ using some block theory. This was later generalized by M. Herzog, using the Theory of blocks with cyclic defect group, to classify finite groups with a cyclic Sylow $p$-subgroup of order $p^{a},$ but which have order less than $p^{3a}.$ Both of these results predate the classification of finite simple groups by some margin. Brauer's classification would suffice to deal with this question.

The only non-Abelian simple groups which appear in Brauer's list are ${\rm PSL}(2,p)$ and ${\rm SL}(2,p-1)$ (the latter only when $p$ is a Fermat prime).