Proof involving group and subgroup

[Deveno]

Hi Arcana!



Slight modification for (iv)->(i):

Ha = Hb \Rightarrow \forall h' \in H, \exists h'' \in H:h'a=h''b \Rightarrow \quad ... \quad \Rightarrow ab^{-1} \in H

That is, start with the condition of (iv), reformulate it as specific elements, and rewrite it to the condition of (i).

what i like to do is think of these things like this:

Ha = Hb. so now we need something "element-wise" involving a and b.

so every h'a in Ha is some h"b in Hb. don't really care which, we're not peeking that far under the hood, so we just pick some h' and h" that makes that true:

h'a = h"b.

now we need to get to something that is of the form:

blah blah blah = ab^-1

to get a all by itself, we want to multiply the equation by something on the left, or otherwise we'll mix up h-somethings with the b (that would be counter-productive). h'^-1 seems like the obvious choice:

a = h'^-1h"b.

now...do you see how to get b^-1 on the left, and take b off the right?

 

[ArcanaNoir]

I'm not going to continue working on this problem.

 

[I like Serena]

Must be something in the air. Perhaps autumn?
You're the second today that's apparently frustrated and says so in the thread.

Ah wait! Today daylight saving switched to wintertime.
Did your clock switch?

 

[Deveno]

well, that's your choice, but:

if at some later date, some theorem uses this fact without further comment in its proof (and i can assure you that will happen), you will find yourself mystified.

these 4 conditions are used interchangeably to identify common cosets of H, and crop up again and again. this problem isn't just "some example to show how groups work", it's part of the "standard vocabulary" of cosets.

so, skipping this problem, it may not negatively affect, say, your grade on this assignment. but it's like deciding "i'm not going to learn how to read words that start with the letter T".

math education is cumulative, if you skip some small parts early on, you have "bigger holes" later.

i mean, look, it's no skin off MY nose if you work through this or not, but i am telling you straight-up, you're selling yourself short. expect a steep uphill climb in the rest of the course.

 

[ArcanaNoir]

well, that's your choice, but:

if at some later date, some theorem uses this fact without further comment in its proof (and i can assure you that will happen), you will find yourself mystified.

these 4 conditions are used interchangeably to identify common cosets of H, and crop up again and again. this problem isn't just "some example to show how groups work", it's part of the "standard vocabulary" of cosets.

so, skipping this problem, it may not negatively affect, say, your grade on this assignment. but it's like deciding "i'm not going to learn how to read words that start with the letter T".

math education is cumulative, if you skip some small parts early on, you have "bigger holes" later.

i mean, look, it's no skin off MY nose if you work through this or not, but i am telling you straight-up, you're selling yourself short. expect a steep uphill climb in the rest of the course.

Gee, is everyone cranky today?
Just because I'm not doing this problem at this time doesn't mean I won't ever learn it. It's just that I have a huge test on Tuesday and many many theorems and exercises to be familiar with and this one is just taking too much time and frustrating me. I'm past the point where I can mentally deal with this problem because my stress level is just too high right now. I do appreciate all the help I've received on this problem, but if I'm going to postpone having a mental breakdown until after the test, then this problem needs to be put on the shelf for now. Yes, it's entirely possible this problem might be on this test, but it also might not, and so in essence I'm sacrificing my potential to get this problem on the test for the greater good of studying several other theorems that are equally likely to be on the test. :) (yes that's the smile of a crazy person trying grasp at the straws of sanity)

 

[Deveno]

Gee, is everyone cranky today?
Just because I'm not doing this problem at this time doesn't mean I won't ever learn it. It's just that I have a huge test on Tuesday and many many theorems and exercises to be familiar with and this one is just taking too much time and frustrating me. I'm past the point where I can mentally deal with this problem because my stress level is just too high right now. I do appreciate all the help I've received on this problem, but if I'm going to postpone having a mental breakdown until after the test, then this problem needs to be put on the shelf for now. Yes, it's entirely possible this problem might be on this test, but it also might not, and so in essence I'm sacrificing my potential to get this problem on the test for the greater good of studying several other theorems that are equally likely to be on the test. :) (yes that's the smile of a crazy person trying grasp at the straws of sanity)

i understand you are stressed out because you have a lot of material to cover, and you don't want to spend a lot of time on "just one problem" that's driving you crazy. and i don't know how much material (how many chapters, concepts, etc.) is going to be covered on this test.

what i would like to point out, is that not all problems are created equal. some are more "disposable" than others. and what I'm trying to communicate to you is that THIS problem, is actually important, you will use this later on (even if it's NOT an important question on your test). i can't tell you if "getting" this exercise wil help you on the test, or not, it may not. but what i CAN say, with certainty, is that if you do not understand this problem inside-out, you will struggle with cosets for...well, at least as long as you're doing group theory.

many textbooks (Dummit and Foote, Herstein, Jacobsen, Artin) will prove something like ab^-1 is in H, and then use the cosets Ha and Hb interchangeably. at some point, you will probably cover the orbit-stabilizer theorem, and this entire problem will be "condensed" in it.

i'm not trying to be an unsymapthetic person. i know what it is like to have x amount of time, and x + y amount of stuff to review (where y is large compared to x). what i am trying to give you some sense of, is that regardless of how well you do on "the test", you need to know this stuff. the object of taking a course, is to acquire the knowledge, not "a good grade in it".

stumbling over basic concepts, in order to do well short-term, isn't a good long-term strategy. you may do fine on the upcoming test, by ignoring this issue, and covering other material you've been ignoring, and i understand the value of that to you. but, in the long-run, i'd rather see you ace the final, or the mid-term, even at the expense of some stuff in-between, rather than start out strong, and get hopelessly stuck at the end.

so...look, i can't tell you what to do, and in the end, it's not really my business. but...if you don't do this sometime, you'll never really LEARN this stuff, and then whatever grade you got, is meaningless, it's just some printing on some paper somewhere.

so...here's a compromise...let it sit a bit, you're obviously tired of banging your head against this wall. take a quick look sometime tomorrow, maybe it will make some more sense after you've relieved some of that stress.

 

[Evo]

The Op has requested a rest for the thread.