Let $a$ and $b$ be elements of a group $G$, and $H$ and $K$ be subgroups of $G$. If $aH=bK$, prove that $H=K$.

Since $ae=a \in aH$, it follows that $a\in bK \implies aK=bK=aH$. Now, multiply by $a^{-1}$ both sides and you have $H=K$

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Group Theory

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