# Recognition of finite simple groups by number of Sylow p-subgroups (2)

This is a follow-up to this question.

Let $G$ and $G'$ be two finite simple groups of the following structures:

1- $A_{p}$, for some primes $p$;

2- $PSL_{p}(q)$, for some prime $p$ and some prime power $q$;

3- $PSU_{p}(q)$, for some prime $p$ and some prime power $q$;

Also suppose that $s$ is a prime number dividing $|G|$ and $|G'|$ and every Sylow $s$-subgroup of $G$ and $G'$ is a cyclic subgroup of order $s$. If $G$ and $G'$ have the same number of Sylow $s$-subgroups, then can we say that $G\cong G'$?

• Note that $p=2$ is allowed in $PSL_{p}(q)$. – H.Shahsavari Jan 15 '18 at 15:29

${\rm PSL}(2,31)$ and ${\rm PSL}(2,32)$ both have $496$ Sylow $3$-subgroups.