= Trivial intersections of Sylow subgroups strengthen the Sylow congruence
{c}
{title2=$n_p\equiv1\pmod{|P|}$}
Suppose distinct <Sylow subgroups> for $p$ in a <finite group> have trivial intersections. Conjugation by one of them, $P$, fixes $P$. The <stabilizer> of any other $Q$ is $H=P\cap N_G(Q)$. Since $H$ normalizes $Q$, the product $HQ$ is a <subgroup> of $p$-power order; maximality of $Q$ gives $H\leq Q$. Thus $H\leq P\cap Q=1$. All other <orbits> have size $|P|$, proving the congruence.
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