Let $p$ and $q$ be prime divisors of finite group $G$. Also let $n_{p}$ be the number of Sylow $p$subgroups of $G$ . Is there any example such that $n_{p}=n_{q}\neq 1$? Thanks in advance.

2$\begingroup$ There are $33$ such groups of order less than $256$. $\endgroup$– Tom De MedtsJul 6 '12 at 16:08

3$\begingroup$ This is hardly a research level question! $\endgroup$– Derek HoltJul 6 '12 at 16:35

1$\begingroup$ Consider any Frobenius group with nilpotent complement not of prime power order. There are many such groups. $\endgroup$– Geoff RobinsonJul 6 '12 at 16:42
Unless I miscomputed, this happens in the group of affine transformations ($x \mapsto ax+b$ with $a\neq 0$) over the field of 7 elements. There seem to be 7 2Sylow subgroups and 7 3Sylow subgroups.

$\begingroup$ This is true in groups of affine transformations mod a a prime $l$ more generally, for $p$ and $q$ any two prime divisors of $l1$. The stabilizer of a $p$ or $q$Sylow subgroup is the stabilizer of the unique point it fixes. $\endgroup$ Jul 6 '12 at 16:33