- #1
Wiemster
- 72
- 0
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:
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: Rn / Zn = (R/Z)n which is supposed to b homomorphic to the unit circle. But I don't really see what this 'factor group' represents... )
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: Rn / Zn = (R/Z)n which is supposed to b homomorphic to the unit circle. But I don't really see what this 'factor group' represents... )