If $G$ is a finite group and $P$ is a Sylow $p$-subgroup of with $P=HK$, $H$,$K$ are subgroups of $P$ and if $Q$ is a Sylow $q$-subgroup of $G$ and $H^{a}Q=QH^{a}$, $K^{b}Q=QK^{b}$ for some $a,b \in G$. Is there any chance that $PQ^{t}=Q^{t}P$ for some $t\in G$?
Sylow subgroups of a group G
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abstract-algebra
group-theory
finite-groups