Coset representatives when a subgroup $H$ is not normal.

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Let $G$ be a group and let $H$ be a subgroup of $G$.

If $H$ is normal then the set of all left coset representatives is the same as the set of all right coset representatives.

If $H$ is not normal, does there exist a set of left coset representatives which happens to also be a right coset representatives?

asked Dec 19, 2019 at 3:18

Rachid Atmai

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$\begingroup$ Regardless of $H$ being normal or not, take $eH=He$, where $e$ is the identity. $\endgroup$
– Anurag A
Dec 19, 2019 at 3:20
$\begingroup$ Not the intended meaning of the question :p Anything aside $\{e\}$ though? $\endgroup$
– Rachid Atmai
Dec 19, 2019 at 3:21
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$\begingroup$ take something in the center of the group if the center is non-trivial. $\endgroup$
– Anurag A
Dec 19, 2019 at 3:22
$\begingroup$ perhaps this will help you: math.stackexchange.com/questions/178186/… $\endgroup$
– Anurag A
Dec 19, 2019 at 3:24
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$\begingroup$ See math.stackexchange.com/questions/134523/… and math.stackexchange.com/questions/3298084/… and math.stackexchange.com/questions/268219/… $\endgroup$
– Gerry Myerson
Dec 19, 2019 at 4:13

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