I need a pretty basic explanation of cosets

[Asmoridin]

So, I'm trying to self-teach myself Abstract Algebra, and this idea of cosets is killing me, and I'm not completely sure why.

Basically, I think I understand the theory, but I'm having a hard time visualizing it. Does anyone know a basic example of a group that would have a different right and left cosets?

Thanks!

 

[micromass]

I understand your trouble. When I first was introduced to quotient groups and cosets, I wanted to visualize it too. But I found out that there simply is no correct way to visualize it...
The only visualization I could make was that a quotient group is setting something equal to zero. But that was it.

So don't feel annoyed because you're unable to visualize cosets...

Anyway, a nice counterexample is found in the dihedral group of order 6, do you know this? Basically, it's the group of automorphisms on a triangle. Its elements are

$$\{e,a,a^2,b,ab,a^2b\}$$

where e is doing nothing, where a and a² are rotations and where the rest are reflections. It's operations are generated by the following relationships

$$a^3=e,~b^2=e,~ab=ba^2$$

Anyway, consider the subgroup {e,b}. It's left cosets are

$$\{e,b\},\{a,ab\},\{a^2,a^2b\}$$

while it's right cosets are

$$\{e,b\},\{a,a^2b\},\{a^2,ab\}$$

So this is an example where the left cosets are not the right cosets...

 

[Asmoridin]

Oh! That is PERFECT. It actually took me a few minutes to work all of that out (*sigh), but that definitely helped clear up a lot of this theory stuff.

So with {e,b}, the index would be |G|/|H| = 3, hence the three cosets (which was something else I just couldn't grasp until now), and this isn't a normal subgroup because they aren't equal- that's perfect!

I really appreciate the help. I'm sure I'll be popping on here, occasionally. I definitely need the help sometimes :(.

Thanks!

 

[micromass]

Oh! That is PERFECT. It actually took me a few minutes to work all of that out (*sigh), but that definitely helped clear up a lot of this theory stuff.

So with {e,b}, the index would be |G|/|H| = 3, hence the three cosets (which was something else I just couldn't grasp until now), and this isn't a normal subgroup because they aren't equal- that's perfect!

I really appreciate the help. I'm sure I'll be popping on here, occasionally. I definitely need the help sometimes :(.

Thanks!

I'm glad you appreciated it And I hope to see you popping up here!

 

[Stephen Tashi]

A slight digression on the topic of visualizing, I used to have thoughts like this:

If G is a group, we may be able to find an interesting set of subsets of G that forms a group under the "natural" definition of a group operation, which would be that if s1 and s2 are in these subset then their product is the set (s1)(s2) = {the set of all elements g such that g = (g1)(g2) where g1 is some element of s1 and g2 is some element of s2. The subsets we use don't have to be mutually exclusive and they don't have to contain the same number of elements.

I mentioned this to one of my professors and he wasn't enthusiastic about it. He felt I was mentally poisoning myself. I wonder why. Is this thought too simplistic? It should have some relation to cosets, shouldn't it?

 

[mathwonk]

i like to visualize cosets via the action of a group on a set. In the example above the dihedral group acts on the vertices of a triangle. If the vertices are X,Y,Z, the subgroup leaving say X fixed, contains two elements, the identity and one reflection. Its left cosets consist of elements that take X to other points. I.e. following any element that fixes X by an element taking X to Y also takes X to Y.

If you think of the rotation taking Y to X, then following such a rotation by an element fixing X will still take Y to X. So the right cosets of the subgroup fixing X, consist of elements taking some other one element to X, while the left cosets consist of elements taking X to some other one element.

 

[mathwonk]

by the way, actions come in two flavors and my explanation was for a left action. also every subgroup H of every group G occurs as the subgroup fixing some element under a left action by G. namely let the group G act on the left cosets of the given subgroup H by left multiplication. then the subgroup H fixes itself, the left coset xH consists of all elements sending H to xH, and the right coset Hx consists of all elements sending x^-1.H to H.

 

[Robert1986]

I am currently in my first graduate-level algebra class. I have taken Abstract Algebra I & II and we used the book Topics In Algebra by Hernstein. I sort of kind of got the whole deal about cosets and quotient groups but I never really, really understood it until this semester. This semester we are using Dummit and Foot's book and on the first page of Chapter 3 is a diagram that a)helped me visualize cosets and b)helped me actually understand much, much more the idea of quotient groups.

 

[gauss^2]

Oh! That is PERFECT. It actually took me a few minutes to work all of that out (*sigh), but that definitely helped clear up a lot of this theory stuff.

So with {e,b}, the index would be |G|/|H| = 3, hence the three cosets (which was something else I just couldn't grasp until now), and this isn't a normal subgroup because they aren't equal- that's perfect!

I really appreciate the help. I'm sure I'll be popping on here, occasionally. I definitely need the help sometimes :(.

Thanks!

Have you seen these videos from Harvard's first abstract algebra class?

http://www.extension.harvard.edu/openlearning/math222/

I have watched the first 7 of them on group theory, and they definitely seem to give some intuition behind the stuff that just seemed like never-ending definition-theorem-proof when I took the subject in undergrad years ago. We used Dummit & Foote, which I didn't feel like I learned a lot from. Reading some of Artin (which is used in the video class above) and doing the problems, I feel Artin is better in the chapters I have read so far (only 1 & 2, so a small sample size I realize). I can't wait to move on to the more geometric part of the book later on (but I want to spend some time working chapter 1 of Lang of first).

 

[Asmoridin]

Have you seen these videos from Harvard's first abstract algebra class?

http://www.extension.harvard.edu/openlearning/math222/

I have watched the first 7 of them on group theory, and they definitely seem to give some intuition behind the stuff that just seemed like never-ending definition-theorem-proof when I took the subject in undergrad years ago. We used Dummit & Foote, which I didn't feel like I learned a lot from. Reading some of Artin (which is used in the video class above) and doing the problems, I feel Artin is better in the chapters I have read so far (only 1 & 2, so a small sample size I realize). I can't wait to move on to the more geometric part of the book later on (but I want to spend some time working chapter 1 of Lang of first).

Videos, you say?

That's certainly worth checking out, I haven't actually seen any lectures on the stuff, but having some would certainly be a great resource- I'll take a look when I get home. I really really appreciate it!