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.

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    $\begingroup$ There are $33$ such groups of order less than $256$. $\endgroup$ Jul 6 '12 at 16:08
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    $\begingroup$ This is hardly a research level question! $\endgroup$
    – Derek Holt
    Jul 6 '12 at 16:35
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    $\begingroup$ Consider any Frobenius group with nilpotent complement not of prime power order. There are many such groups. $\endgroup$ Jul 6 '12 at 16:42

Unless I mis-computed, 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 2-Sylow subgroups and 7 3-Sylow 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 $l-1$. The stabilizer of a $p$ or $q$-Sylow subgroup is the stabilizer of the unique point it fixes. $\endgroup$
    – Will Sawin
    Jul 6 '12 at 16:33

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