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 $t\in G$?
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Well there may be a chance that it's true, but it is not always true! Let $G=A_5$, $P \in {\rm Syl}_2(G)$, $Q \in {\rm Syl}_5(G)$. Then $P \cong C_2 \times C_2$, so $P=HK$ with $|H|=|K|=2$. All elements (and hence also all subgroups) of order 2 in $G$ are conjugate, and there exist subgroups of $G$ of order 10, so it is true that $H^aQ=QH^a$, $K^bQ=QK^b$ for some $a,b \in G$, but $G$ has no subgroup of order 20, so it is not true that $PQ^t=Q^tP$ for some $t \in G$. |
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