Visual Group Theory, Lecture 4.2: Kernels

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Visual Group Theory, Lecture 4.2: Kernels

The kernel of a homomorphism is the set of elements that get mapped to the identity. We show that it is always a normal subgroup of the domain, and that the preimages of the other elements are its cosets. This means that we can always quotient out by the kernel, and this key observation leads us to the fundamental homomorphism theorem. We concluding with two visual examples: one using multiplication tables, and the other using Cayley diagrams.

  1. Professor Macauley
  2. Mathematics
  3. Group theory
  4. Visual group theory
  5. Homomorphism
  6. Kernel
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Комментарии
Автор

truly fantastic, thank you for uploading

zpie
Автор

Thanks for preparing and posting these videos. I've been studying group theory for years; nevertheless, I'd not seen a presentation of group theory in terms of Cayley diagrams, so while watching your videos, I'm still learning something new. It's wonderful to have these new perspectives / "languages" with which to study group theory.

kevinbyrne
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Nobody:
Millennials in Textmessages: 30:33

em-ildq
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May I point out two small issues with your sketch of proving that the preimage of h is a subset of gK? (1) you never posit that phi(g) = h, so there is not enough info to prove g' and g are in the same coset, and (2) at the end, g'=gk should be phrased as g' = gx and the reader should determine that x is in K.

rasraster
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In the proof, weren't you also required to show that if phi(a) <> phi(b) then a <> b?

rasraster