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Let $H$ be a subgroup of $G$. I'm asked for which $g \in G$ is $Hg := \{hg \vert h \in H\}$ is a subgroup. Obviously it is for $g \in H$.

I'm not sure about the other cases. I can't seem to find another example, since there would have to be an $h \in H$ so that $hg = e$, but this would imply $h = g^{-1}$ which contradicts $g$ not being in $H$. Is there anything I missed?

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    $\begingroup$ nope, you've proved that $H$ is the only right coset of $H$ which is also a subgroup - or indeed $Hg$ is a subgroup if and only if $g\in H$. $\endgroup$ – Robert Chamberlain Apr 22 '18 at 19:12
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Consider $g\in H$ such that $Hg$ is a subgroup. This means $e_G\in Hg$. There exists $h\in H$ such that $hg=e_G$ i.e. $g=h^{—1}\in H$.

It is obvious that when $g\in H$ one has $Hg=H$ that is a subgroup. So the only coset $Hg$ that is a subgroup is $H$

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