Visual Group Theory, Lecture 3.5: Quotient groups

Visual Group Theory, Lecture 3.5: Quotient groups
Like how a direct product can be thought of as a way to "multiply" two groups, a quotient is a way to "divide" a group by one of its subgroups. We start by defining this in terms of collapsing Cayley diagrams, until we get a conjecture about what property a subgroup H needs to have for the quotient G/H to exist. At this point, we translate everything into formal algebraic language and prove this theorem. Specifically, the quotient exists when the set of left cosets of H forms a group. This requires a well-defined binary operation, which exists if and only if H is a normal subgroup.
Course webpage (with lecture notes, HW, etc.): www.math.clemson.edu/~macaule/...

This one is 40 minutes!?!? Things about to get real.

[@speedbird7587]

It is really strange that not all abelian subgroups of a group are normal !

[@speedbird7587]

You really explain the topic amazingly. Thanks you.

[@hanyanglee9018]

3:11 for someone who loved the bubbles 4 years ago, your dear bubbles are back.

[@rickyda10]

Dear Professor , I am not sure but I think in slide #7 the colors of the generators 'a' and 'x' should be interchanged in the description above the picture. Otherwise I don't understand it anymore.. Thankyou for this video serie, it really helps me to understand Group theory.

[@Mrpallekuling]

Yes, in the picture, a blue arrow goes from e to x, so x is to be blue. A green arrow goes from r to z so that is green as shown. Thiis means a is to be red.

[@schweinmachtbree1013]

just pointing out that the associative property wasn't checked in the slide at 35:35 (although this is probably because it is very easy and wouldn't have fit on the slide lol)

[@Harsimran_Singh_27]

What is V-4 in this video?

[@scitwi9164]

Something is not right here: If a direct product joins *corresponding* nodes in each subgroup, then why in the *quotient* should we be allowed to merge nodes which are _not corresponding_? (as if the example with the blue lines crossing over) I don't see it as being legit, and the fact that we cannot restore the original group by product again might be thought of as a confirmation of that :P So why are those "invalid" quotients called "quotients" too? Shouldn't the term "quotient" be reserved only to those cases where we can get the actual _factors_ back? If this gives something which is not a factor, then I would say it shouldn't be called "quotient" in the first place, but perhaps something else (similarly as left cosets _look like_ subgroups, but they _aren't_ ones because they don't contain the identity element). Is there any rationale behind this confusing definition of a quotient? Because from what I see, it can produce more problems than it solves, and might lead to unnecessary confusion.

[@TheHarvardOpportunes]

I think this problem you're talking about is something that makes the quotient really interesting! Remember the quotient is all about "collapsing", and making groups simpler - like if we see a bunch of normal subgroups, we want to collapse them into single nodes because it can make underlying properties of the group easier to understand. Whenever we make things simpler, we're definitely gonna lose some information, but we can get a lot of benefits out of it as well. In your example, if we cared too much about blue lines crossing, we'd be missing out on the really cool (and most of the time more important) fact that the blue lines map us perfectly from one subgroup to another subgroup - we don't care about the distinctions between corresponding elements anymore because we've just collapsed them to make things simpler! Also if it helps, remember that we could have re-drawn the cayley diagram to show the correspondences you're looking for, just at the expense of the structure of the red arrows.

[@hanyanglee9018]

Yeah, when quotient doesn't reverse the product, things get wired. But notice that, the difference between "original" and "divided and then multiplied" could probably be some clue for us to build up giant groups, if the fact of no reversible quotient could be found from the bigger result group does count. Imo, A4 is some kinda base which works like the prime numbers. It's not dividable somehow. But, this is tutorial about groups. Groups are naturally more complex than numbers. It shouldn't be hard to imagine that group has more than one ways to be divided. I think the quotient introduced in this video is only one of them. If you prefer, you could define some. Like, let's say, some manipulation makes C2 to C6, the reverse one makes C6 back to C2, and this looks very much alike some dividing intuitively. Actually I've no idea what the quotient hides. I'm too new in this area. The answer might be in the videos after this one.

[@hanyanglee9018]

27:05 Btw, I think even if H dot H is not defined to be H itself, it still is. H is a group, so it's closure. If H is denoted in the form of set, it's very clear. H = {e, generator1, gen2, gen3, ...| some properties} or { e, gen1, gen2, gen1 gen2, gen3, ...}. Then H dot H is also a set. After simplification, it results back into H itself. So the definition is unnecessary.

[@speedbird7587]

Dividing a group by two isomorphic normal subgroups does not necessarily result in two isomorphic quotients! I am really interested to know what is the necessary and sufficient conditions for the two quotients be isomorphic?

[@Agus-of6rh]

Lost it from slide 6 and on. It's not clear why the Cayley diagram of A4 is that one. This is neither specified in previous videos.

[@reinerwilhelms-tricarico344]

How to get from the diagram at 13:30 to the next one at 13:42 really doesn't occur to me. I don't have the foggiest idea.

[@Mrpallekuling]

The short answer is that one generator is added and the nodes are re-arranged, but it takes quite some time to verify.

[@rasraster]

31:25 is actually not obvious. I.e., putting parentheses around b1H and then replacing that by b2H. Your coset multiplication was simply presented as a definition, and it's not at all clear that your defined operation allows for one of the product's factors to be joined to H and manipulated independently of a1.

[@rasraster]

Went digging around and found a clear element-wise proof in Fraleigh book, Theorem 14.4.

[@xanderlewis]

I'm not sure what you think isn't obvious. b1H and b2H are the same coset, so they can be interchanged freely. Am I misunderstanding you?

[@evankim4096]

@@xanderlewis they are the same coset because they were defined to be in the same equivalence class [b]

[@kittyworldgamingcompany]

and "manipulated independently" is allowed because the group operation is associative

[@qingzhenwu8370]

@3:30 arent that 3+H? if 1+H i thought the bottom should be 1---3---5

[@Mrpallekuling]

The text says 1+H=1, 3, 5 so the illustration should be in the same order (1, 3, 5) to avoid confusion. It does not matter for the final result, but it's annoying IMHO.

[@justinshin2279]

Really appreciate the hard work that went into these lectures but I have to say that for someone new to group theory the reliance on Cayley graphs is probably really confusing. And by that I mean I’m confused and I’ve gone through abstract algebra class a few times...

[@atzuras]

I have a new theorem for you all:: you cannot do proper math without 2 color pens.

[@mathematicia]

Counter example : if you have enough imagination then you can only do it be one pen or no pen

[@isaacdarche7103]

The word is un-does. Un-dos is not a word.

[@dispatch1347]

As a lecturer and one that's tried to make videos before, it's hard to give a 40 minute lecture without messing up a single word.