let $G$ be a finite non-abelian simple group.If there exist $p$ and $q$ which are different prime numbers of $|G|$ such that $n\_p(G)=n\_q(G)$?



Question on the equal Sylow number in finite non-abelian simple group
Not sure if this question really qualifies for MO.


Anyway, the answer very much depends on the group $G$. In most cases $n\_p(G)\ne n\_q(G)$ for distinct prime divisors of the group order. However, there are infinitely many examples where equality occurs: If $r$ is an odd prime, then $n\_p(\text{PSL}(2,r))=r(r+1)/2$ for each odd prime divisor $p$ of $r-1$.


But there are other examples too. For instance the atlas of finite simple groups shows that in the Janko group $J\_1$, the normalizers of the $3$-Sylows and $5$-Sylows have order $60$.


