WEBVTT
Kind: captions
Language: en
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We are back now say as we would recalled,
last time we were discussing Cayley graphs
00:00:21.919 --> 00:00:31.571
of groups. So, what has given to me is a group
and generating set for this group. So, the
00:00:31.571 --> 00:00:37.300
case of our fourth we had taken in generating
said to be r and f, r as we would remember,
00:00:37.300 --> 00:00:46.409
it was a rotation by pi by 2 and f was flipping
about x axis.
00:00:46.409 --> 00:00:56.989
And then we could see that the Cayley graph
of D 4 is cubic. I had put two conditions,
00:00:56.989 --> 00:01:04.820
these two conditions. So, the vertices of
the Cayley graph, they were points of the
00:01:04.820 --> 00:01:11.250
group, there were elements on the group and
then two elements two nodes two vertices of
00:01:11.250 --> 00:01:20.800
the graph were connected by an edge if either
a minus b, but an S or b inverse a or in S.
00:01:20.800 --> 00:01:27.740
What is the use of expressing groups in this
fashion. Many applications would come and
00:01:27.740 --> 00:01:36.890
before that let us practice more.
Let us see in few more examples , very simple
00:01:36.890 --> 00:01:51.150
example I would take Z 6 the cyclic group
of order 6, what are the elements, I will
00:01:51.150 --> 00:02:04.190
just write them as 1 a a square a cube
a raise to the power of 4 a raise to the power
00:02:04.190 --> 00:02:11.250
5 a raise to the power 6, then a raised to
power 6 e is identity.
00:02:11.250 --> 00:02:25.349
So, better I will not write this only first
6 elements I am going to write this much.
00:02:25.349 --> 00:02:42.819
I would have also written this as zero, 1,
2, 3, 4, 5, these 5 symbols and these are
00:02:42.819 --> 00:02:53.790
forming a group under addition modulo 6. So,
these are the remainders and the addition
00:02:53.790 --> 00:03:07.359
modulo 6. So, 5 plus 4 therefore, is 3, because
that is what happens, mod 6. So, what about
00:03:07.359 --> 00:03:16.599
the generating set.
So, the cyclic group. Cyclic group meaning
00:03:16.599 --> 00:03:21.569
there is one element in this group which is
capable of generating it entirely. So, the
00:03:21.569 --> 00:03:28.430
way I have written like this, its very clear
that is a one of them. Well I could have taken
00:03:28.430 --> 00:03:36.819
a to the power 5 as then, a to the power 5
would have been as good as a, because a to
00:03:36.819 --> 00:03:46.489
the power of 5 and then a to the power 5 to
the power 2, which is a power 10, a to the
00:03:46.489 --> 00:03:53.680
power 15, a to the power 20 and a to the power
25; that actually is same as this, because
00:03:53.680 --> 00:03:58.010
a to the power 5, a to the power 6 as a we
call a is 1.
00:03:58.010 --> 00:04:07.379
So, it your 25 is therefore, just a, because
a to the power 24 is identity and now this
00:04:07.379 --> 00:04:19.780
is a square, this is a cube a to the power
10 is a to the power 4 is a. So, all the elements
00:04:19.780 --> 00:04:30.759
are there. So, a as well as a to the power
5, both of both the, both elements are generators.
00:04:30.759 --> 00:04:42.900
Is there any other generating set. Well I
would say a 2 together with a 3 also generates
00:04:42.900 --> 00:04:57.460
the same thing.
So, if I take S to be equal to a square a
00:04:57.460 --> 00:05:10.580
3. Oh I just take it to be 2 3; that is a
generating set for cyclic group as the cyclic
00:05:10.580 --> 00:05:24.960
group Z 6 . So, there are two options let
me take as to be a, then how does the Cayley
00:05:24.960 --> 00:05:38.860
graph look like, I have 1, I have a, I have
a cubed, I have a raised to the power, sorry
00:05:38.860 --> 00:05:55.240
this was a square .
So, this was a square a cube a raise to the
00:05:55.240 --> 00:06:08.090
power 4 a raise to the power 5 like this and
then do I connect a raise to the power 5 with
00:06:08.090 --> 00:06:19.460
1. So, what is the condition for connecting
edges. So, either a inverse b should belong
00:06:19.460 --> 00:06:30.460
to S or b inverse a should belong to S.
So, if I take this to be a, this to be b,
00:06:30.460 --> 00:06:40.000
a inverse b, a is a which actually belongs
to, was a to the power 5 inverse 1; that is
00:06:40.000 --> 00:06:51.490
a and that belongs to S right.
So, I will connect it back. So, the name is
00:06:51.490 --> 00:07:02.939
quite justified cyclic group. I have connected
say a to a square, because a inverse times
00:07:02.939 --> 00:07:16.169
a square is a which belongs to this set. So,
what about some other generating set. So,
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I take this to be generating side, this time
this one, this generating set and . So, let
00:07:25.240 --> 00:07:53.300
us see what happens. So, I have identity
and then I have a square, I have a cubed and
00:07:53.300 --> 00:08:23.240
then back to this and then I have oh. Sorry
a square I have a to the power 4 and then
00:08:23.240 --> 00:08:27.160
back to this, I am connecting it, because
a square belongs to.
00:08:27.160 --> 00:08:36.141
So, here I am doing a inverse b. Let us a
square which belongs to S generating set and
00:08:36.141 --> 00:08:58.820
now I have a cube here and then a raise to
the power 5 and then a raise to your 7 and
00:08:58.820 --> 00:09:06.630
back to this, and then I also connect a square
with a 5, because a to the power minus 2 times
00:09:06.630 --> 00:09:13.810
a to the power 5 which is a cube, actually
belongs to S and similarly I do like this.
00:09:13.810 --> 00:09:19.401
So, it turns out to be the, Cayley graph turns
out to be a triangle in another triangle and
00:09:19.401 --> 00:09:30.510
all these vertices getting connected by an
edge to the outer void x, do the some other
00:09:30.510 --> 00:09:39.840
group of order 6 Z sis is 1 another group
is S 3 which is the group of permutations
00:09:39.840 --> 00:09:47.310
of three elements.
So, with that group also, let us do an experiment.
00:09:47.310 --> 00:09:59.910
Remember this, remember this picture and do
the experiment with S 3. So, what is S 3 or
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call it D 3. You want to understand it as
understand it as as S 3, then this is group
00:10:13.980 --> 00:10:37.180
of permutations of three symbols , will not
understand it as D 3, then this is group of
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symmetries of three gon an equilateral triangle.
So, we called what is D 3 D n we had seen.
00:11:03.950 --> 00:11:24.830
So, we have identity, we have flipping,
we have rotation by this case 2 pi by 3
00:11:24.830 --> 00:11:42.020
1 20 degree and then we have f r we have r
squared and we have f r square and its not
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very difficult to see that f r is actually
same as r square f and f r square is same
00:11:48.330 --> 00:11:54.690
as r f.
So, this is, this 6 elements are there f if
00:11:54.690 --> 00:12:10.800
you want. So, here is a triangle f, if you
want is flipping about y axis and r is rotation
00:12:10.800 --> 00:12:24.420
by 2 pi by 3. So, what would would be the
Cayley graph. So, I take generating set as
00:12:24.420 --> 00:12:35.330
two elements r and f, r and f action during
the whole, whole D 3. So, I have identity
00:12:35.330 --> 00:12:50.600
element and then I have r, then I have r square
and then again r square will be connected
00:12:50.600 --> 00:13:03.870
to 1, because a inverse b. So, r square
inverse 1 which is just r r belongs to S.
00:13:03.870 --> 00:13:13.180
So, I make these connections and then similarly
I have smaller triangle here inside. So, I
00:13:13.180 --> 00:13:29.250
have r and this is f. So, this is r f and
this is f. So, this is f here, and here I
00:13:29.250 --> 00:13:40.040
have r square f. So, the graph turns out to
be like this. So, you carefully look at this
00:13:40.040 --> 00:13:54.530
graph and carefully look at the graph that
was their f or the case of Z 3 Z 6, what the
00:13:54.530 --> 00:14:00.860
graphs look same are these two groups same.
No these two groups are not same; one is a
00:14:00.860 --> 00:14:05.320
billion group, another is nonbillion group.
So, these are two different groups what their
00:14:05.320 --> 00:14:09.140
graphs look quite.
Similar you see there is a triangle line that
00:14:09.140 --> 00:14:14.380
is trying it outside and then these vertices
are inner vertex is connected to the outer
00:14:14.380 --> 00:14:26.060
one and similarly f or the group D 3, you
have same thing coming out. Is there a way
00:14:26.060 --> 00:14:33.250
to distinguish these two. Can you read from
the graph somewhat like this is a graph of
00:14:33.250 --> 00:14:43.600
D 3 and the other one is the graph of
graph of Z 6. So, here is a very nice way,
00:14:43.600 --> 00:14:52.190
which is you, you give it direction, you consider
directed graphs.
00:14:52.190 --> 00:15:08.710
So, you can consider directed graphs. In fact,
you can also have directed colored graphs.
00:15:08.710 --> 00:15:15.750
So, you can give some direction in that picture,
you can also put some colors in that picture,
00:15:15.750 --> 00:15:21.210
so that you have a way to understand which
one is as 3 and which one is for a billion
00:15:21.210 --> 00:15:31.750
group of 6 order; So, how do I put the
graph, how to, how do you put the direction
00:15:31.750 --> 00:15:53.310
to this graph. So, I first do it for D 3 for
which I know generating set is f and r what
00:15:53.310 --> 00:15:59.940
are the elements 1.
So, first maybe I will , I will put some.
00:15:59.940 --> 00:16:11.810
So, maybe I will put some vertices first
and then I am going to connect them with edges.
00:16:11.810 --> 00:16:28.732
So, 1 r r square and here is my scheme, I
would put this thing. So, this is a, this
00:16:28.732 --> 00:16:43.570
is b. I would put arrow like this if a inverse
b belongs to S that is it. So, if b inverse
00:16:43.570 --> 00:16:59.360
a belongs to S, in that case I would put.
So, it b inverse a belongs to S, then I would
00:16:59.360 --> 00:17:17.039
put a picture like this and if both a
inverse b then b inverse a, if both of them
00:17:17.039 --> 00:17:26.990
belong to S and then I would put picture like
this, then I will give direction to this what
00:17:26.990 --> 00:17:33.070
I can also do is, I can actually color my,
I can color code my generators. So, for each
00:17:33.070 --> 00:17:43.620
generator you have a color.
So, here let me just fix one color. For r
00:17:43.620 --> 00:17:52.890
and we do have this color. For f, I am going
to have this color. So, I am going to have
00:17:52.890 --> 00:18:03.740
colored and directed, directed by . So, direction
I would good, I would put using these conditions
00:18:03.740 --> 00:18:09.250
and color I anyway put to all the generators.
So, how many colors are there, say as many
00:18:09.250 --> 00:18:15.650
as number of generators and remember each
choice of generating set gives you a different
00:18:15.650 --> 00:18:41.030
Cayley graph ok.
So, first I am going to just put the vertices
00:18:41.030 --> 00:19:12.050
1 r r square and f r f r square. So, these
are vertices and how to put color in the direction.
00:19:12.050 --> 00:19:24.350
So, first I am going to do it for r. So, one
I would connect to r 1 inverse r is belonging
00:19:24.350 --> 00:19:31.910
to S, S, it is belonging to S and the direction
is therefore, this.
00:19:31.910 --> 00:19:39.000
And now I am going to put the direction like
this, because r inverse times r squared is
00:19:39.000 --> 00:19:52.500
r which belongs to S and here I am going to
put a direction like this. So, you have 1
00:19:52.500 --> 00:20:04.520
r r square r cube is same as 1 ok what about
f f, we have chosen green color. So, the color
00:20:04.520 --> 00:20:15.650
the direction would be both sides, because
f is same as f inverse. So, f as well as
00:20:15.650 --> 00:20:24.059
f inverse both belong to S.
Now, f is here, what about this f is here
00:20:24.059 --> 00:20:40.870
r f is there, do I connect them when let us
see. So, f inverse r f. So, here its f it
00:20:40.870 --> 00:20:48.000
is r f. So, should I join them. So, in order
to realize whether I should join the invert
00:20:48.000 --> 00:20:53.590
to with direction, I should consider r , I
should consider f inverse r f and r f inverse
00:20:53.590 --> 00:21:04.690
f. So, what is our f inverse, let us see.
So, this is f inverse r inverse take something
00:21:04.690 --> 00:21:10.090
interesting, whenever you have two elements
x and y in a group.
00:21:10.090 --> 00:21:17.520
So if you multiply these two group these two
elements in a group, take the inverse, it
00:21:17.520 --> 00:21:30.720
is actually y inverse x inverse and this property
has a very curious name, which is called socks
00:21:30.720 --> 00:21:43.020
shoe property quite curiously. So, using socks
shoe property I conclude that this is f inverse
00:21:43.020 --> 00:22:08.390
r inverse f and this is f r square f . So,
what would it be? So, I i said some time ago
00:22:08.390 --> 00:22:17.470
that r is square same is f r. So, I have f
and this is f r and f square is 1. So, this
00:22:17.470 --> 00:22:29.059
is just r which actually belongs to S and
what about this , again this is f r f and
00:22:29.059 --> 00:22:39.690
as I computed here r f is same as r r f is
same as f r square. So, this is f f r square.
00:22:39.690 --> 00:22:44.010
So, this is r square which does not belong
to us
00:22:44.010 --> 00:22:52.200
So, here direction should be from r f to f
there, actually should be this and similarly
00:22:52.200 --> 00:23:03.360
if I do it for others direction, would be
this and here the direction would be this
00:23:03.360 --> 00:23:14.559
and here r f and r. So, I have r inverse r
f which is f, which belongs to S right. So,
00:23:14.559 --> 00:23:22.760
from r to r f, I will put direction like this
what about r f inverse r are you socks are
00:23:22.760 --> 00:23:29.430
you socks shoe property. So, this is f inverse
r inverse r which is f, which I mean belongs
00:23:29.430 --> 00:23:36.550
to S. So, I have both the directions.
Similarly, here also I have both the directions.
00:23:36.550 --> 00:23:45.620
So, as far as directed graph of D 3 is concerned,
it is this. So, this is colored as well as
00:23:45.620 --> 00:23:59.350
directed. So in fact, using directed colored
graph I can ,
00:23:59.350 --> 00:24:15.440
I can very its very easy for me to read
relations in the group how is that. So, in
00:24:15.440 --> 00:24:22.990
order to reach this point, for example, what
you do, you take r and then you take f, this
00:24:22.990 --> 00:24:37.650
green was f, take r f that is all or you do
f r r f r square. So, r f is same as f r square
00:24:37.650 --> 00:24:46.460
other relations also you can see within this.
So, here you see the direction of the outer
00:24:46.460 --> 00:24:52.700
triangle is like this, which is anti clockwise,
but for the inner triangle the direction is
00:24:52.700 --> 00:25:07.780
like this, which is clockwise what about
the direction of, what would be directed color
00:25:07.780 --> 00:25:19.620
the graph for Z 6 you have to think about
it, maybe that is an exercise for you and
00:25:19.620 --> 00:25:27.570
that will make you realize that the directed
color graphs for Z 6 and directed telegraph
00:25:27.570 --> 00:25:37.630
for D 3.
They are actually different, quite often all
00:25:37.630 --> 00:26:02.110
these graphs are often studied as geometric
objects and we are not going to be bothered
00:26:02.110 --> 00:26:11.310
with that in this course, but to understand
groups people use geometry and the branch
00:26:11.310 --> 00:26:26.140
of mathematics, which is called geometric
group theory, but we are not going to talk
00:26:26.140 --> 00:26:35.610
about geometric group theory in these lectures.
Let us see one more interesting graph in this
00:26:35.610 --> 00:26:55.500
discussion which is a graph of f 2. What is
f 2 free group on two generators .
00:26:55.500 --> 00:27:01.290
Do you remember free groups on two generators
or n generators, there are two symbols a and
00:27:01.290 --> 00:27:10.760
b and we make words of all possible lines
out of these two alphabets and we also put
00:27:10.760 --> 00:27:17.330
two more alphabets a dash and b dash and we
use all four alphabets to make words we make
00:27:17.330 --> 00:27:19.010
dictionary.
And then there are some equivalent words,
00:27:19.010 --> 00:27:24.910
collection of all those words under juxtaposition,
its just back to back we just put those two
00:27:24.910 --> 00:27:31.410
words and form a new word that is called;
that is a group operation for free groups.
00:27:31.410 --> 00:27:45.310
So, how to draw the graph Cayley for f 2.
So, I take f a be free group on two generators
00:27:45.310 --> 00:27:57.490
a b, you just naming them as a b, what are
the relations? There are no relations and
00:27:57.490 --> 00:28:04.200
since there are no relations, there are no
circuits, there are no closed cubes in the
00:28:04.200 --> 00:28:27.370
graph. So, no relations means no closed cubes
in the graph, how does it work.
00:28:27.370 --> 00:28:38.760
So, let me take. Now I am taking color coded
thing, let me take this as a and let me take
00:28:38.760 --> 00:28:57.750
this as b so that. So, I am taking like this
and then the reverse direction I have a dash
00:28:57.750 --> 00:29:10.420
and I have b dash, rather I would use b vertically,
I need horizontally just a, just a way to
00:29:10.420 --> 00:29:22.809
avoid relations. So, then there as far as
possible, so I have
00:29:22.809 --> 00:29:48.980
so I have a, then I have b. So, this is a
b, this point is a b or I have here inverse
00:29:48.980 --> 00:30:04.660
and then I have here b. So, this is a, a inverse
b inverse a inverse b and here again if I
00:30:04.660 --> 00:30:20.080
take a. So, I take, if I can do a, I can do
a inverse, I can also do b or I can come back
00:30:20.080 --> 00:30:24.510
here.
So, like that I have and here again I have
00:30:24.510 --> 00:30:35.720
options of going up, coming down, going right,
going left and similarly at this point also
00:30:35.720 --> 00:30:43.510
I have the option of doing this, option of
doing this, an option of doing this. And once
00:30:43.510 --> 00:30:51.809
we reach these these points again I have
option to do this, to do this. Sorry the horizontal,
00:30:51.809 --> 00:31:05.330
the vertical will be green. So, to do this
and to do this and to do this, and similarly
00:31:05.330 --> 00:31:12.850
here I have option 2 at this point, I have
option to do this or I have option to do this,
00:31:12.850 --> 00:31:23.040
I have option to do this. Be careful, be careful
here no crossings .
00:31:23.040 --> 00:31:37.170
So, eventually what you get its, its an image
which you may like to call your, heard of
00:31:37.170 --> 00:31:49.260
this word fractal, its a huge graph, it is
a graph with infinitely many vertices. So,
00:31:49.260 --> 00:31:56.179
many vertices that is quite, quite big and
you cannot see relations in this, because
00:31:56.179 --> 00:32:04.540
there are no close the group. The group is
free group , what what is the application,
00:32:04.540 --> 00:32:09.830
what is the use of having all these graphs
Cayley graphs
00:32:09.830 --> 00:32:18.440
We are going to talk about all this in coming
lectures when the interesting application
00:32:18.440 --> 00:32:23.520
would come in understanding puzzles, in understanding
Rubik's cube. In fact, whenever there is a
00:32:23.520 --> 00:32:31.070
group action you can construct a graph out
of it, and some more graphs we are going to
00:32:31.070 --> 00:32:41.120
join coming lectures, and lots of those graphs
are going to come from symmetry. We are going
00:32:41.120 --> 00:32:50.390
to understand symmetry groups, groups of symmetries
of various geometric objects, various things
00:32:50.390 --> 00:32:59.380
like this you recognize what it is.
Every side of this is triangle and then
00:32:59.380 --> 00:33:09.530
there are 1 2 3 4 vertices, its quite familiar
I guess you saw it in your chemistry course,
00:33:09.530 --> 00:33:17.840
its tetrahedron tetra stands for 4, hedron
stands for number of faces, it has 4 faces.
00:33:17.840 --> 00:33:22.270
So, tetrahedron we are going to understand
symmetries of this. As I told symmetry is
00:33:22.270 --> 00:33:25.429
going to be one of the important aspects of
this course.
00:33:25.429 --> 00:33:31.140
So, symmetry is what we are going to understand
and if you try to write Cayley graph of
00:33:31.140 --> 00:33:40.309
this, you can try to write the colorfully
directed graph of this and all the action,
00:33:40.309 --> 00:33:46.840
all the movements of this we can understand
through the movements on the graph. So, to
00:33:46.840 --> 00:33:52.470
know more about these things, this is just
one example of what are called platonic solids.
00:33:52.470 --> 00:33:56.970
So, to understand all this, keep watching
I will see you next time