Timeline for remark in milne's class field theory notes
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2010 at 16:56 | comment | added | Pete L. Clark | No, that's not true either: one does not need the structure theorem for finite abelian groups in order to prove this. See e.g. math.uga.edu/~pete/4400algebra2point5.pdf, where the canonical isomorphism of an abelian group with its second dual is proved in Section 4 and the structure theorem, which lies deeper, is not touched until Section 5. (I mention this for your edification, not to beat you down.) | |
| Jan 28, 2010 at 11:05 | comment | added | Harry Gindi | I acknowledge my error and I should have added that the groups need to be abelian. Also, the standard proof for finite abelian groups uses the structure theorem at least once to reduce to the case of cyclic groups, so the proof of finite case doesn't generalize well to the general case. | |
| Jan 28, 2010 at 8:53 | comment | added | Pete L. Clark | A canonical isomorphism with the double dual holds for all locally compact abelian groups. | |
| Jan 28, 2010 at 8:43 | comment | added | Harry Gindi | To add to Ben's statement: However, in the case of the pontryagin double-dual (of a finite group), a canonical isomorphism does exist to the double-dual. This is because there is a canonical injection into the double-dual, and by proving that the group is isomorphic to its dual, you also prove an isomorphism between the dual and the double dual. However, in the case of finite groups, every injective homomorphism between isomorphic groups is in fact an isomorphism. This is exactly the same proof as for finite dimensional vector spaces. | |
| Jan 28, 2010 at 8:42 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
added 35 characters in body
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| Jan 28, 2010 at 8:27 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |