Group theory 17: Finite abelian groups

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This lecture is part of a mathematics course on group theory. It shows that every finitely generated abelian group is a sum of cyclic groups.

Correction: At 9:22 the generators should be g, h+ng not g, g+nh
  1. Richard E Borcherds
  2. finite abelian group
  3. finitely generated abelian group
  4. group theory
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Комментарии
Автор

If I'm not mistaken at 9:22, the change in generators should be from {g, h} to {g + nh, h} because if we change to {g, g + nh} instead we may not be able to recover h as we are working over a ring and n may not be an unit.

shivajidas
Автор

Thank you so much for these priceless lectures again. These are fantastic for reference for understanding what’s actually going on on the subject, as some of the proofs in textbook often doesn’t give a terribly useful insight for what the theorems are actually doing other than accomplishing the result. This theorem is definitely one of them. I’m was reading this in Hungerfold lately and it was proved inductively non-constructively (for my poor brain) and I was so confused by the mysterious dividing chain condition and how it should work in practice. The proof given in this video just answered all of my questions about this theorem.

gunhasirac
Автор

Finally something I seem to be able to absorb with no pain at all:
Every finite abelian group is the product (but he said sum) of prime cyclic groups.
This proves I know SOMETHING, but also suggests that what I do know is not much.

RalphDratman
Автор

How do you know you have enough non trivial relations?

vikramsundara