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

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 $$9$$.

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.
• Thank you very much! That is very helpful. By the way, Is there a classification of finite simple groups with cyclic Sylow $p$-subgroup (of order larger than $p$)? Commented Feb 1 at 8:56
• For every Lie type there are many. Random example: ${^3}\!D_4(19)$ at the prime $13$. Commented Feb 1 at 9:42
• In this example, $19^4-19^2+1$ is divisible by $13^2$ but no other irreducible factor in the group order formula is divisible by $13$. This gives you a clue as to how to find many such examples. The same happens for $89^4-89^2+1$, as a random example. Commented Feb 1 at 10:02
• If you want a more extreme example, $\operatorname{\rm PSL}(3,2819)$ has a cyclic T.I. Sylow $19$-subgroup of order $19^4$. Commented Feb 1 at 11:50