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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 $G\cong G'$?

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  • $\begingroup$ Note that $p=2$ is allowed in $PSL_{p}(q)$. $\endgroup$ – H.Shahsavari Jan 15 '18 at 15:29
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${\rm PSL}(2,31)$ and ${\rm PSL}(2,32)$ both have $496$ Sylow $3$-subgroups.

I think that when you ask question like this, you should provide at least some kind of reason, however tenuous, as to why you might expect the answer to be yes. Conjectures in group theory that are made purely on the basis that one is unable to think of a counterexample do not usually turn out to be correct.

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