Classification of non-abelian simple groups with cyclic T.I. Sylow p -subgroup

Question

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is a trivial intersection (for short T.I.) subgroup of $G$ if $H\cap H^x=1$ for each $x\in G-N_G(H)$.

I read the next result in the paper The p-local ranks of finite simple groups with abelian Sylow p-subgroups.

Theorem Let $G$ be a non-abelian simple group with a non-cyclic T.I. Sylow $p$-subgroup $P$. Then $G$ is isomorphic to one of the following groups:

(a) $\mathrm{PSL}_2(q)$, where $q=p^n$ and $n \geq 2$;

(b) $\operatorname{PSU}_3(q^2)$, where $q^2=p^n$;

(c) $p=2$ and $G \cong{ }^2 B_2(2^{2 m+1})$;

(d) $p=3$ and $G \cong{ }^2 G_2(3^{2 m+1})$ and $m \geq 1$;

(e) $p=3$ and $G \cong \mathrm{PSL}_3(4)$ or $M_{11}$;

(f) $p=5$ and $G \cong { }^2 F_4(2)^{\prime}$ or $McL$;

(g) $p=11$ and $G \cong J_4$.

I am wondering whether there is a similar version for non-abelian simple groups with a cyclic T.I. Sylow $p$-subgroup?

Note that the class of non-abelian simple groups with $p$-part $p$ is one of the classes. But, this is not the only one. For instance, $\mathrm{PSL}_2(17)$ and $\mathrm{PSL}_3(8)$ both have a cyclic T.I. Sylow $3$-subgroup of order $9$.

Answer 1

This is answered in the paper of Harvey Blau, "On Trivial Intersection of Cyclic Sylow Subgroups" Proc AMS 1985. Whenever a Sylow $p$-subgroup of a finite simple group is cyclic, it is T.I. The proof uses the classification, of course. The proof shows that if a finite simple group has a cyclic Sylow $p$-subgroup of order greater than $p$ then the group has Lie type with characteristic not equal to $p$. There are very many like this.