Visual Group Theory, Lecture 3.2: Cosets

Visual Group Theory, Lecture 3.2: Cosets
The "regularity" property of Cayley diagrams implies that identical copies of the fragment corresponding to a subgroup appear throughout the rest of the diagram. These subsets are called cosets. In this lecture, we formalize this algebraically and prove some basic properties about them. There is a natural notion of left coset and right coset, and these are frequently different. We analyze this both algebraically and in terms of Cayley diagrams. Finally, we conclude with an important theorem due to Joseph Lagrange, relating the size of a subgroup to the number of its cosets and the size of the original group.
Course webpage (with lecture notes, HW, etc.): www.math.clemso...

[@turbolesso]

It's the best thing in the world, when you understand theorems and concepts in math, which your prof couldn't teach you in a simple way. Thanks a lot for your effort. Keep up the great work. I will binge watch this, till the explosion of my mind.

So nice to have an intuitive idea of what a coset is!

[@scitwi9164]

So if the subgroup and its cosets partition the group into identical subsets, can we say that subgroups are like divisors and cosets are like their multiples? As for the integers example: so the cosets of the subgroup of multiples of 4 are all the particular residue classes modulo 4? (E.g. all integers with the remainder of 1 are in one coset, all integers with the remainder of 2 are in the second coset, all integers with the remainder of 3 are in the third coset, and all integers with the remainder of 0 are in the original subgroup.) Am I right? Regarding the Lagrange's Teorem: Ha! Looks like I _was_ right after all! It's all about divisors! :) Which means that if I happened to live before Lagrange, this would be now called Sci Twi's theorem :D

[@kevinbyrne4538]

You're more correct than you realize. If a group G has a (normal) subgroup H, then a new group (the quotient or factor group) can be defined, with H as its identity element and the cosets as the other elements of the group. This group's operation is the combination of the cosets according to the rule: aH * bH = abH . The new group is symbolized G/H (i.e., as a quotient, since H divides G into cosets that have an equal number of elements, just as dividing a set of, say, 12 eggs by 4 means separating the 12 eggs into 4 sets of 3 eggs each.)

[@amon1him]

Thank you so much for these amazing videos!

[@sasha-2574]

very awesome! Thank you!

[@solid7965]

Why do you prefer to write fr as r^2f? Seems a little bit arbitrary, and warrants some clarification to the casual observer. (Since I'd prefer to write anything, and just have it accepted, but can't seem to get away with that)

[@ir6plans60]

18:52 is this proof really sufficient shouldn't we prove that all of the left cosets of subgroup have the same size?

[@hellow4164]

You are right, not sure if it was proved somewhere else. Specifially, you don’t just want to show cosets are all the same size but that they are exactly the size of the subgroup H. You can do this by showing that the aH is a bijective mapping from h to ah (if you can’t see why immediately try writing out a proof for it), then you can proceed with the rest of the proof of Lagrange’s Theorem.

[@ronycb7168]

Thanks Sir Liked Bsc PCM lol 😂

[@fsaldan1]

Small error in the last sentence. It should be the order of G divided by the order oH and not the other way around. Misspoke.