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It's not much, but I'm still a little bit confused about cosets, and I noticed that

$xH = H$ implies $x$ to be an element of $H$.

Why does this make sense? Thank you!

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If $H$ is a subgroup, then the identity element $e\in H$, so $x=xe\in xH=H$.

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\begin{alignat}{1} &H=xH \Longrightarrow \\ &H\subseteq xH \Longrightarrow \\ &\forall h \in H, \exists h'\in H \mid h=xh' \Longrightarrow \\ &\forall h \in H, \exists h'\in H \mid x=hh'^{-1}\Longrightarrow \\ &x\in H \end{alignat}

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