3

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?

Rachid Atmai
  • 1,152
  • Regardless of $H$ being normal or not, take $eH=He$, where $e$ is the identity. – Anurag A Dec 19 '19 at 03:20
  • Not the intended meaning of the question :p Anything aside ${e}$ though? – Rachid Atmai Dec 19 '19 at 03:21
  • 4
    take something in the center of the group if the center is non-trivial. – Anurag A Dec 19 '19 at 03:22
  • perhaps this will help you: https://math.stackexchange.com/questions/178186/relation-between-left-and-right-coset-representatives-of-a-subgroup?rq=1 – Anurag A Dec 19 '19 at 03:24
  • Thank you! Qiaochu Yuan seems to hint at an answer in the comments. – Rachid Atmai Dec 19 '19 at 03:28
  • I think Hall's Marriage Theorem tells you that the set of left cosets and the set of right cosets have a common transversal, a common set of distinct representatives. Indeed, I think it was in this context that Hall first came up with the theorem. – Gerry Myerson Dec 19 '19 at 04:09
  • 1
    See https://math.stackexchange.com/questions/134523/left-and-right-transversals-of-groups and https://math.stackexchange.com/questions/3298084/left-transversal-of-finite-group-that-is-also-right-transversal and https://math.stackexchange.com/questions/268219/mutual-set-of-representatives-for-left-and-right-cosets-what-about-infinite-gro – Gerry Myerson Dec 19 '19 at 04:13
  • Gerry, thank you for rephrasing the problem and the links. I had never heard of the notion of transversal. It basically a choice set. Thanks Anurag A. for the help! – Rachid Atmai Dec 19 '19 at 06:57

0 Answers0