Group theory 4: Lagrange's theorem

This is lecture 4 of an online course on mathematical group theory.
It introduces Lagrange's theorem that the order of a subgroup divides the order of a group, and uses it to show that all groups of prime order are cyclic, and to prove Fermat's theorem and Euler's theorem.

[AndrewLi]

This is the first time I've seen the use of orbits for the discussion of cosets. Really appreciate it!

[julesjacobs]

Your lectures are amazing! I love that you explain the big picture.

[jackozeehakkjuz]

At 25:00 it should be 2¹¹-2 is divisible by 11.

[sgssergio]

The numbers of cosets is called index too and denoted by [G:H]?

[netrapture]

I like the check on coset disjointness starting at 15:35, but couldn't we say at 16:07 "for some h_1, h_2" rather than for "some g_1, g_2" and at 16:09 we could say "show g_1.H = g_2.H" rather than "assuming g_1.H = g_2.H"

[PunmasterSTP]

Coset? More like "Cool information with which we'll be all set!" 👍

[mikeywatts]

At 6:30, why is the identity element necessarily in H? That is, if G acts on a set S, then what makes 1s=s? I guess, what does it mean for a group G to act on a set S? I interpret that to mean an arbitrary function f: (G x S) -> S. I guess I'm misinterpreting the definition of "acts on"?

[dipanjanpal]

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[shaneyaw]

This lecture was poorly planned and executed. The video neglects important topics, hardly clarifies abstract concepts, and the written notes are almost incomprehensible. It would have been worthwhile for the lecturer to slow down and clearly define the mathematical concepts, especially if mathematical rigor is to be expected from the student.