GT3. Cosets and Lagrange's Theorem

Abstract Algebra: Let G be a group with subgroup H. We define an equivalence relation on G that partitions G into left cosets. We use this partition to prove Lagrange's Theorem and its corollary.

[charlesbrowne]

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[juancamilolopez]

Hi! I´m from Colombia, i´m studying math at Andes university. I like very much your videos! Thank you and keep doing this :)

[yotube]

you do Abstract Another year with Dr. Bob!!

[MathDoctorBob]

So the statement is that two two classes are disjoint or equal. This amount to the following - if x is in Ey, then Ex=Ey. Ex subset Ey - xRy by assumption. If z is in Ex, then zRx. By transitivity, zRy and y is in Ey. Likewise for the other direction. Hope that helps!

[MathDoctorBob]

They will always wind up evenly spaced, although that needs a proof. For the example you list, closure means we need to include 1 + 1 = 2, 1 + 1 + 1 = 3, ... and we get everything. If n is 1 or is coprime to 10, n will generate all of Z/10. - Bob

[MathDoctorBob]

Each cycle is shorthand. (12) means 1->2, 2->1, 3->3, and (123) means 1->2, 2->3, 3->1. To multiply read cycle right to left, so (12)(123) means 1->2->1, 2->3->3, 3->1->2 or (23).

[theultimatereductionist]

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[MathDoctorBob]

Order will depend on the group operation, which means we need to be careful when groups have addition and multiplication. In Z/10 with addition, 5 + 5 = 0, so the order of 5 is 2 under addition. Under multiplication, Z/10 is not a group (why?), so instead we use the multiplicative units (Z/10)* = 1, 3, 7, 9. I say more about these in the Aut(Z/n) video.

[MathDoctorBob]

Thanks! Group theory is where I live.

[MathDoctorBob]

@nahaymath My guiding philosophy is to look as close to TV or a PowerPoint presentation as possible. Lighting is a huge factor (two light boxes from Adorama in NYC), and I prefer a whiteboard (black on white) to most electronic notepads (white on black). The black T-shirt is more than a fashion choice.

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[MathDoctorBob]

:) You're welcome! Also check out the course on UReddit, which has problems sets with solutions.

[MathDoctorBob]

@themonstera The equivalence relation is on G, so we are comparing elements x, y of G. I'll annotate to make clearer. Thanks! - Bob

[MathDoctorBob]

It actually begins with the Math Major Basics playlist (logic, proofs, sets). I'm working on Ring Theory notes now.

[umanicole]

Thanks Dr. Bob! I like that triangle for S_3. Super-helpful! My Alg Final is Thursday! Wish me luck! I'm almost done with my 😊

[kevinbyrne]

6:20 -- Why does (12) x (123) = (23) ?  Draw a triangle.  Label the vertices (clockwise) 1, 2, 3.  Now apply the permutation (123):  1 --> 2, 2 --> 3, 3 --> 1.  Now apply the permutation (12) ; this exchanges the labels of what were initially labeled vertices 1 and 2.  Since 3 is now sitting at the vertex that was originally labeled 1, then 3 is moved to the vertex that was originally labeled 2.  Since 1 is now sitting at the vertex that was originally labeled 2, then 1 is moved to the vertex that was originally labeled 1.  However, (12) doesn't affect the label at the vertex that was originally labeled 3 ; since that vertex is now labeled 2, then 2 isn't moved by (12).  The next result of (12) x (123) is that 1 has been returned to its original position, but 2 and 3 have exchanged positions, so (12) x (123) = (23).

[nidepngyou]

Freaking beautiful.
I love this. I love you.
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[Steve]

Good lecture. Is there a problem set (homework) to go with it?

[MathDoctorBob]

That's graduate material (manifolds), and, while it is in my specialty, I don't see getting around to it for a long while.

[skndestroy]

In 14:01 when you are chosing your subgroups H, you picked {0, 5} and {0, 2, 4, 6, 8}.. could you have also picked the subgroup with elements {0, 1, 5, 6, 9} which has order 5 OR do your elements have to be evenly spaced as in your examples?

[BlackMarketHero]

why does (1 2 3) x (1 2) go to (2 3), i don't see how it went from 3 elements to 2