My lecture notes state that every element of a group belongs to exactly one right and one left coset of a certain subgroup, but I don't see why this should be the case, so I tries to prove it:(adsbygoogle = window.adsbygoogle || []).push({});

with S and S' elements of the subgroup H of G we then have that if a certain element X of the group G belongs to the coset TH but to T'H as well

i.e. X=TS=T'S' we should have that T=T' right?

This means TS(S'^-1)=T'E with S(S'^-1) again an element of H, but I don't see why T should be T', or did I misintepreted the theorem?

(PS: Somewhat further they use as an example a generelaized torus:R^{n}/Z^{n}= (R/Z)^{n}which is supposed to b homomorphic to the unit circle. But I don't really see what this 'factor group' represents... )

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# Elements in only 1 left/right coset

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