25:22 I think the important thing about the elements in the subgroup is that they commute with every element of the group. (These are the only two elements with this property).
"At the top, you will notice that the same cosets come back" The fact that the operation is commutative does guarantee that this will happen. However, you can still get this effect when the operation is not commutative. When it happens, your initial subgroup is called "normal".
Yes, when a group is commutative, you always get the same blocks. But even for non-commutative groups you sometimes still get the same blocks. The symmetries of the square are a good example where this happens (see later in the video @23:23). The answer given by @TheOiseau is absolutely correct 🐦
25:37 isn't that just orientation again but both are split into 2 by the modulo 2 of the amount of times rotated? 0 and 180 have mod 0, 90 and 270 have mod 1. edit: better way to put it is that all the elements in the subset/cosets can reach one another by rotations of 180 degrees c:
At 15:00 i was confused. Under which operator is the quotient group defined? How can we say that it is definitely a group if there's no operation associated with it? Are we showing this purely from it's cayley table?
Thanks for the excellent question. You're right that our explanation focuses on the elements of the group and not the operation. You can simply "pull" the original operation up to the level of the new group. If you take any representatives x and y from the 2 cosets, you can multiply them using the original operator. Their product xy is in a third coset. The key point is that this new coset does not depend on your choice of representatives. And now you just think of this as a new kind of "multiplication" directly on the cosets. I hope this answers your question. Either way, keep asking great questions like this!
@@AllAnglesMathah because whatever I use as the initial subgroup will become the identity of the larger quotient group once I've made the rest of it. For a concrete example: we picked a subgroup 0,4,8 to start with, and then we generated the coset 1,5,9 using 5 as a representative. Since this coset has members 5+{members of 0,4,8 subgroup}, it is tautological that adding a member of the 0,4,8 subgroup to a member of this coset will always return another element in that coset. Hence collectively, elements of {048} act like an identity element on {159} (adding them does nothing, in the sense that it will not move us to a coset with a different representative) Then I can intuit that it didn't matter that I picked 5. Furthermore, I can intuit that perhaps if I added members from two non-identity cosets, it would always give me elements from another unique coset. Also I can understand that the order I created my cosets in doesn't matter, because I can always rearrange my rows and columns in the Cayley table to make all the different cases the same. Also, the video explains that the representatives I chose for each coset don't matter. So no choice that I actually made when making the Cayley table at 15:00 actually matters. The quotient group's structure is something intrinsic to the original group, and all I have done is rearranged things to make it more obvious. Am I on the right lines? Edit: sorry, I've said subgroup lots of times when I should've said coset, fixed now hopefully
Nice video! Around 14:56 , did you mean to write Q=G/S? I'm used to colon being used for the index, but maybe there's some regional variation (since colons are used for proportions, etc.).
I used the colon notation without much thought. Your comment made me check a few sources, and it seems that the slash is indeed much more common in group theory. Too late to fix the video, but thanks for pointing this out.
honey wake up new allangles video
This is honestly one of the most creative compliments we've had 😄
25:22 I think the important thing about the elements in the subgroup is that they commute with every element of the group. (These are the only two elements with this property).
Wow, I hadn't noticed that yet. This is quite an advanced way of looking at it. Nice!
That’s the center, right? And since the center is a normal subgroup, it can form a quotient group
@@enpeacemusic192 I think that's correct, yes. But never take my word for it, I'm not an expert.
Indeed, mathematics IS the ultimate natural science, the queen of them all.
Amazing! Can't imagine how much effort you put for making these educational videos, please keep running this channel.
Thank you! We'll keep making videos as long as we can 🙂
Your explanation with animations is very good, every day I become more fascinated by mathematics
13:40 - is that because our operation is commutative?
"At the top, you will notice that the same cosets come back"
The fact that the operation is commutative does guarantee that this will happen. However, you can still get this effect when the operation is not commutative. When it happens, your initial subgroup is called "normal".
Yes, when a group is commutative, you always get the same blocks. But even for non-commutative groups you sometimes still get the same blocks. The symmetries of the square are a good example where this happens (see later in the video @23:23).
The answer given by @TheOiseau is absolutely correct 🐦
@@TheOiseau non-trivial normal subgroups my beloved
Thank you!
25:37 isn't that just orientation again but both are split into 2 by the modulo 2 of the amount of times rotated? 0 and 180 have mod 0, 90 and 270 have mod 1.
edit: better way to put it is that all the elements in the subset/cosets can reach one another by rotations of 180 degrees c:
That's a nice and interesting way of looking at it. So the main concept to take away from this, would be 180 degree rotational symmetry.
At 15:00 i was confused. Under which operator is the quotient group defined? How can we say that it is definitely a group if there's no operation associated with it? Are we showing this purely from it's cayley table?
Thanks for the excellent question. You're right that our explanation focuses on the elements of the group and not the operation.
You can simply "pull" the original operation up to the level of the new group. If you take any representatives x and y from the 2 cosets, you can multiply them using the original operator. Their product xy is in a third coset. The key point is that this new coset does not depend on your choice of representatives. And now you just think of this as a new kind of "multiplication" directly on the cosets.
I hope this answers your question. Either way, keep asking great questions like this!
@@AllAnglesMathah because whatever I use as the initial subgroup will become the identity of the larger quotient group once I've made the rest of it.
For a concrete example: we picked a subgroup 0,4,8 to start with, and then we generated the coset 1,5,9 using 5 as a representative.
Since this coset has members 5+{members of 0,4,8 subgroup}, it is tautological that adding a member of the 0,4,8 subgroup to a member of this coset will always return another element in that coset.
Hence collectively, elements of {048} act like an identity element on {159} (adding them does nothing, in the sense that it will not move us to a coset with a different representative)
Then I can intuit that it didn't matter that I picked 5. Furthermore, I can intuit that perhaps if I added members from two non-identity cosets, it would always give me elements from another unique coset. Also I can understand that the order I created my cosets in doesn't matter, because I can always rearrange my rows and columns in the Cayley table to make all the different cases the same. Also, the video explains that the representatives I chose for each coset don't matter.
So no choice that I actually made when making the Cayley table at 15:00 actually matters. The quotient group's structure is something intrinsic to the original group, and all I have done is rearranged things to make it more obvious.
Am I on the right lines?
Edit: sorry, I've said subgroup lots of times when I should've said coset, fixed now hopefully
@@itoastpotatoes399 Yes, you're getting the correct line of reasoning!
Nice video! Around 14:56 , did you mean to write Q=G/S? I'm used to colon being used for the index, but maybe there's some regional variation (since colons are used for proportions, etc.).
Some people use : for division too. Like ÷ without the line.
I’ve seen him use “:” for “/“ in previous videos, so I think it was intentional but you understood correctly
I used the colon notation without much thought. Your comment made me check a few sources, and it seems that the slash is indeed much more common in group theory. Too late to fix the video, but thanks for pointing this out.