Visual Group Theory, Lecture 3.1: Subgroups

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Visual Group Theory, Lecture 3.1: Subgroups

In this lecture, we begin by examining a property about Cayley graphs called "regularity" that we've hinted at but not yet spelled out explicitly. Next, we introduce the concept of a subgroup, provide some examples, and show how the subgroups of any group can be arranged in a lattice.

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

I don't think that `<=` and `<` are supposed to mean the same. First is for subgroups (which can be of the same size as the original group), and the other is for _proper_ subgroups (which are strictly smaller). A group can be a subgroup of itself, but it is not a _proper_ subgroup.
I have a question though: Is there a formula for the number of all subgroups of a given group? Or a better algorithm of finding all subgroups? From what I see, it can be related in some way to prime factorization and all possible divisors.

scitwi
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G is <= G, but G is not < G. That's a pretty big difference. Otherwise your subgroup lattice at 12:05 would be infinitely recursive.

ernestkirstein
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At 10:40 one of the Cayley's diagram of Z_6 coincides with one of the Cayley diagram of D_3. So they are isomorphic, but Z_6 is cyclic and D_3 is not. Where am I wrong?

laflaca