WEBVTT
Kind: captions
Language: en
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Hello, so during previous lectures, we have
seen a couple of things . We saw Rubik's group,
00:00:29.080 --> 00:00:46.760
we saw symmetries of wallpaper . So, what
was Rubik's group, it was just group of all
00:00:46.760 --> 00:00:54.670
permutations, all possible states of Rubik's
cube . So, in that sense, Rubik's group or
00:00:54.670 --> 00:01:02.210
the Rubik's cube is very tangible representation
of a group of Rubik's cube, you can touch
00:01:02.210 --> 00:01:07.600
it, you can feel it, and group is right there
in your hand, all possible 4 into 10 to the
00:01:07.600 --> 00:01:14.520
power 19 states, more than that, they are
just in your hand right, or the group of symmetries
00:01:14.520 --> 00:01:26.230
of wallpaper . Again this picture is representing
something it is representing a group . This
00:01:26.230 --> 00:01:33.900
toy is again representing something that is
representing a group, so that is something
00:01:33.900 --> 00:01:49.470
.
So, Rubik's group is a tangible object
00:01:49.470 --> 00:02:05.149
that occurs as a group naturally. So, one
of course, has to understand what is the meaning
00:02:05.149 --> 00:02:12.150
of occurs, occurs in the sense that the permutations,
which are there all possible permutations,
00:02:12.150 --> 00:02:19.760
they form a group, but in one single toy,
you can see all possible permutations . Similarly,
00:02:19.760 --> 00:02:27.989
symmetries of wallpapers given a wallpaper
pattern given say one tile, you have so many
00:02:27.989 --> 00:02:37.810
symmetries.
So, for example, rotation, reflection, and
00:02:37.810 --> 00:02:49.200
other things that I mentioned last time, glide
reflection all these things are there, and
00:02:49.200 --> 00:02:58.329
just through one picture you can understand
about a group . So, these are concrete
00:02:58.329 --> 00:03:15.319
examples of group. So, concrete objects, these
are concrete objects that represent a group,
00:03:15.319 --> 00:03:29.060
so concrete objects there. And group as we
know is an abstract structure .
00:03:29.060 --> 00:03:39.780
Through these examples, through the example
of wallpaper symmetry, through the example
00:03:39.780 --> 00:03:45.530
of a Rubik's group that abstract group gets
a meaning, it gets it gets represented in
00:03:45.530 --> 00:03:53.010
a where if you say, real life situation
. So, for us it is always good, if you could
00:03:53.010 --> 00:04:00.030
represent a given abstract group, in some
meaningful fashion . So, here the meaningful
00:04:00.030 --> 00:04:17.720
fashion was permutation, and here the meaningful
representation was all this rotation,
00:04:17.720 --> 00:04:22.370
reflection, glide reflection, all these things
.
00:04:22.370 --> 00:04:29.960
So, here is something quite interesting . Any
abstract group can be understood in terms
00:04:29.960 --> 00:04:37.330
of permutations, any abstract group can be
understood terms of matrices, I hope you recall
00:04:37.330 --> 00:04:43.630
that or rotation and reflection, yesterday
we are talking about matrices . So, these
00:04:43.630 --> 00:04:50.590
two key words, permutations and matrices,
they are quite concrete right. And the statement
00:04:50.590 --> 00:04:58.340
is that any abstract group, what is abstract
group, any group that satisfies all those
00:04:58.340 --> 00:05:05.990
properties in the definition of group, associated
key existence of identity, existence of inverse
00:05:05.990 --> 00:05:08.419
.
So, any abstract group actually can be thought
00:05:08.419 --> 00:05:15.930
of as group of permutations, any abstract
group can be thought of as group of matrices,
00:05:15.930 --> 00:05:26.960
and that is quite relieving that is quite
interesting thing . So, statement is, when
00:05:26.960 --> 00:05:42.669
you have an abstract group, then it can be
thought of as if it is sitting inside S n
00:05:42.669 --> 00:06:03.160
a suitable permutation group, and that is
called Calyley's theorem .
00:06:03.160 --> 00:06:12.409
So, what is that how can we realize . And
in group as as if it is sitting inside,
00:06:12.409 --> 00:06:25.069
S n . So, here is this idea you have a group,
and to this group I want to associate some
00:06:25.069 --> 00:06:34.099
permutation S n. So, first question, what
n, should we take, so here I take n, which
00:06:34.099 --> 00:06:41.810
is same as the cardinality of G, number of
elements of G .
00:06:41.810 --> 00:06:54.940
So, here I am taking finite group. So, here
in this previous slide, it should have been
00:06:54.940 --> 00:07:00.860
finite group . All the for infinite groups
also one can makes in your statement, but
00:07:00.860 --> 00:07:07.150
the actually finite, because you want to get
S n . So, how do I associate a permutation
00:07:07.150 --> 00:07:12.910
to an abstract element to a given element
of group g that is not very difficult, let
00:07:12.910 --> 00:07:23.330
me just say that sigma goes g goes to a permutation
sigma g, a permutation is of n symbols .
00:07:23.330 --> 00:07:30.050
So, what do I do how do I define sigma. So,
sigma g is going to be a permutation that
00:07:30.050 --> 00:07:39.970
is a bijection on G. And how do you give bijection
on group using an element, well one way is
00:07:39.970 --> 00:07:48.870
you just permute in this fashion, you just
multiply g from the left side, and it is clear
00:07:48.870 --> 00:08:05.740
that as a set is set map, sigma g is a bijection
. How do we see that is just cancellation
00:08:05.740 --> 00:08:17.879
property, just left cancellation . So, if
I have a sigma g say h 1, and sigma g h 2,
00:08:17.879 --> 00:08:28.870
that means I am having g h 1, g h 2, both
are equal. Existence of inverse allows me
00:08:28.870 --> 00:08:38.380
to cancel these, so g inverse g h 1 is same
as g inverse g h 2 . And therefore h 1 is
00:08:38.380 --> 00:08:46.110
h 2, and that means, sigma g is a one-one
map .
00:08:46.110 --> 00:08:56.180
So, here is a one-one map, which goes from
a set of finite set to itself . So, it has
00:08:56.180 --> 00:09:07.190
to be onto as well, there is a one-one onto
map, that means, the permutation right, any
00:09:07.190 --> 00:09:12.890
one-one onto map. So bijection, bijection
precisely permutation . So, this sigma g is
00:09:12.890 --> 00:09:16.070
actually permutation.
And you can see that this is a group of a
00:09:16.070 --> 00:09:37.680
morphism, this kind of assignment is a
group homomorphism, not very difficult
00:09:37.680 --> 00:09:52.060
to see . So, therefore, this is a this
is a homomorphism, and I I came that this
00:09:52.060 --> 00:10:06.950
is a injection. So, why should it be injection.
So, just check that if g 1 is so if sigma
00:10:06.950 --> 00:10:18.900
g 1 is same as sigma g 2, then you just need
to check that g 1 is equal to g 2 that is
00:10:18.900 --> 00:10:26.570
not very difficult to make out, so that's
all. So, what is the conclusion, conclusion
00:10:26.570 --> 00:10:31.330
is that every group, every abstract group
can be thought of as a group of permutations.
00:10:31.330 --> 00:10:37.710
So, secondly, for n equal to size of g you
can, but then in some cases, you can also
00:10:37.710 --> 00:10:44.540
optimize this, you can also reduce this n
. So, this is just one algorithm, one way
00:10:44.540 --> 00:10:51.930
to see group as a subgroup of permutations.
So, everything therefore, can be thought of
00:10:51.930 --> 00:11:07.600
as permutation, so that is the message. Message
is every element of a group can be thought
00:11:07.600 --> 00:11:27.040
of as a permutation .
And now, I am making another statement namely,
00:11:27.040 --> 00:11:37.880
every element of a group can be thought of
as a matrix. So, how to see that not very
00:11:37.880 --> 00:12:09.209
difficult . So, statement is every element
of a group, can be thought of as a matrix
00:12:09.209 --> 00:12:15.610
. Not very difficult when even think of
an element as a permutation, you can also
00:12:15.610 --> 00:12:36.250
think of it as permutation matrix .
So, what is permutation matrix. Well you take
00:12:36.250 --> 00:12:53.231
identity matrix, just I am giving an illustration
of 3 by 3 case, and then what do you do,
00:12:53.231 --> 00:13:01.410
just permute rows or columns. So, here let
us permute rows, so I have this is 0 1 0,
00:13:01.410 --> 00:13:25.990
0 0 1, 1 0 0 . So, what happens here . If
I multiply this matrix to x axis, so I am
00:13:25.990 --> 00:13:38.350
taking x, y, z, three axis. So x axis corresponds
to 1 0 0, what happens you know, so when I
00:13:38.350 --> 00:13:46.790
multiply this, I get 0, I multiply this, I
get 0, I multiply this 1 last one, I get
00:13:46.790 --> 00:14:09.660
1 . So, here x axis goes to z axis, what happens,
when I do the same calculation for y axis
00:14:09.660 --> 00:14:29.510
. So, x axis goes to z axis, and for y axis
what happens, so I get 1 here, and I get 0
00:14:29.510 --> 00:14:38.330
here, I get 0 here, so matrix multiplication.
So, y axis goes to x axis.
00:14:38.330 --> 00:14:55.550
And what about z axis, 0 1 0, 0 0 1, 1 0 0,
so I am looking at z axis, 0 0 1, and when
00:14:55.550 --> 00:15:08.700
I multiply what I get, I get 0 here, I get
1 here, and I get 0 here. So, z axis goes
00:15:08.700 --> 00:15:19.910
to y axis. So, z axis goes to y axis. So,
it is a permutation, x goes to z goes to y.
00:15:19.910 --> 00:15:33.310
So, x z y, and then y comes back to x. So,
this matrix is actually representing a permutation.
00:15:33.310 --> 00:15:41.970
What is happening, first row first row of
this is going to z last row, third second
00:15:41.970 --> 00:15:53.080
row is going to first row, and then third
row is going to second row, third row is going
00:15:53.080 --> 00:15:57.310
to second row, it is z is going to y. So,
this matrix is actually representing a permutation.
00:15:57.310 --> 00:16:05.080
So, these such matrices, which are obtained
after permuting rows or say columns of identity
00:16:05.080 --> 00:16:22.880
matrix, they are called permutation matrices.
So, they are obtained by permuting rows or
00:16:22.880 --> 00:16:38.180
columns of identity matrix. So, you can apply
a given permutation on rows or columns.
00:16:38.180 --> 00:16:52.200
So, here is thing, I can therefore, give a
map from S n to group of matrices over a field
00:16:52.200 --> 00:17:03.390
say complex numbers. So, C or C is the set
of a field of complex numbers . And G L n
00:17:03.390 --> 00:17:35.529
C is n by n matrices, with which are invertible
. Those n by n matrices, which are invertible,
00:17:35.529 --> 00:17:39.750
and it forms a group . So, all permutation
matrices being permutation of identity
00:17:39.750 --> 00:17:46.340
matrix. So, permutation of rows or columns,
they determinant is either plus 1 or minus
00:17:46.340 --> 00:17:53.000
1, because you will be doing that permutation
like swapping of rows or columns, either
00:17:53.000 --> 00:18:01.070
even number of times or odd number of times.
So, since the determinant is 1, matrix is
00:18:01.070 --> 00:18:06.629
invertible.
So, any sigma, any permutation can be performed
00:18:06.629 --> 00:18:14.800
on say rows of identity matrix, and therefore
you get let me just call it M sigma, M sigma
00:18:14.800 --> 00:18:37.480
is corresponding permutation matrix. And therefore,
this map is there, and you have to check that
00:18:37.480 --> 00:18:54.540
this map this map is a homomorphism . This
is a homomorphism, you have to check, whether
00:18:54.540 --> 00:19:02.440
it is homomorphism. When sigma is permuted,
sigma permutes n identity element as columns
00:19:02.440 --> 00:19:11.629
or rows, so that is the question ok. So, for
what notion of permutation matrices, whether
00:19:11.629 --> 00:19:15.950
it is in terms of rows, or in terms of columns,
this is a homomorphism nevertheless.
00:19:15.950 --> 00:19:25.919
So, you have a homomorphism from here to here,
it is actually injected homomorphism . So,
00:19:25.919 --> 00:19:37.309
any group sits inside S n, in S n sits inside
G L n over a field, say complex numbers. So,
00:19:37.309 --> 00:19:43.570
therefore, every abstract group can be thought
of as a permutation, and every abstract group
00:19:43.570 --> 00:19:50.100
can also be thought of as a matrix. So, matrices
and permutations, they are very concrete
00:19:50.100 --> 00:19:57.610
examples of elements of a group, so so that
is how these things are useful . So, I
00:19:57.610 --> 00:20:02.690
am representing every abstract element as
a permutation, I am representing every abstract
00:20:02.690 --> 00:20:11.210
element as a matrix, so that is quite possible
.
00:20:11.210 --> 00:20:20.960
So, with that what becomes important are matrix
groups .
00:20:20.960 --> 00:20:26.260
What is the advantage of considering elements
as the matrices, elements of abstract group
00:20:26.260 --> 00:20:35.370
as matrices. When I have an abstract group,
there is no structure to this element G, G
00:20:35.370 --> 00:20:55.909
is atomic. But, when I think of a matrix,
so this is abstract, this is matrix . I can
00:20:55.909 --> 00:21:00.970
perform various operations of matrix itself.
For example, I can compute its determinant,
00:21:00.970 --> 00:21:07.559
I can compute its trace, I can add up all
the entries of it, so many things that I can
00:21:07.559 --> 00:21:13.640
do with matrix, but within abstract element
you cannot.
00:21:13.640 --> 00:21:20.470
So, in some cases, it might be useful to think
of an abstract group as actually matrix
00:21:20.470 --> 00:21:25.850
groups. So, what are the examples of matrix
groups . There are plenty of them, and in
00:21:25.850 --> 00:21:43.270
fact, it is a whole theory of matrix groups,
they can be thought of as what are called
00:21:43.270 --> 00:21:51.610
continuous groups, or say lie groups, or in
terms of algebraic geometry, or in terms
00:21:51.610 --> 00:22:03.190
of polynomial operations, you want to understand
them, what are called algebraic groups, these
00:22:03.190 --> 00:22:08.740
are very very rich theory of mathematics having
their own special case in mathematics.
00:22:08.740 --> 00:22:12.980
We are not going to talk about them, but we
are going to just give an give some examples
00:22:12.980 --> 00:22:20.530
of linear groups.
So, what are the examples, I am saying first
00:22:20.530 --> 00:22:38.809
example that you already know of G L n C,
this called general linear group of order
00:22:38.809 --> 00:22:52.950
n. And then S L n C, what is the definition
of S L n C, is those matrices, those elements
00:22:52.950 --> 00:23:08.809
in general linear group, whose determinant
is 1 . So, elements have determinant 1 form
00:23:08.809 --> 00:23:16.429
a group. So, you take 2 2 matrices, whose
determinant is 1, the product is again determinant
00:23:16.429 --> 00:23:23.259
1, and inverse of matrix whose determinant
is 1 is again having determinant 1, so that
00:23:23.259 --> 00:23:36.290
forms a group called special linear group
.
00:23:36.290 --> 00:23:48.549
Few more groups, this group is O n C. What
is this group, this group is all those matrices
00:23:48.549 --> 00:24:12.440
in G L n C, for which M M transpose is identity
matrix n by n identity matrix. So, here is
00:24:12.440 --> 00:24:25.840
an observation about such elements . Observation
is that if you pick an element from O n C,
00:24:25.840 --> 00:24:40.289
then its determinant is either plus 1 or minus
1, this is because determinant of transpose
00:24:40.289 --> 00:24:49.909
of matrix is same as determinant of matrix,
and therefore determinant of M square is just
00:24:49.909 --> 00:24:58.389
1 ok.
So, this group is
00:24:58.389 --> 00:25:16.619
called orthogonal group. And in fact, there
is a geometric interpretation of this, in
00:25:16.619 --> 00:25:33.619
terms of isometrics . I am not going to
say all this in detail, but this group
00:25:33.619 --> 00:25:44.580
can also be thought of and in fact, general
version of this is closely related to what
00:25:44.580 --> 00:26:01.789
is called isometric group for quadratic forms
. It is quite interesting object has its own
00:26:01.789 --> 00:26:07.370
theory ok.
So, elements of M elements of O n C elements
00:26:07.370 --> 00:26:12.519
of orthogonal group, therefore can be categorized
in two types, the ones which have determinant
00:26:12.519 --> 00:26:19.100
plus 1, and once which have determinant minus
1. So, the ones, which have determinant plus
00:26:19.100 --> 00:26:24.920
1 are called special the collection is called
special orthogonal group.
00:26:24.920 --> 00:26:44.549
So, S O n c, this is all those elements,
in O n, for which determinant, determinant
00:26:44.549 --> 00:27:11.149
is 1, and this is special orthogonal group
There is a reason behind calling them special,
00:27:11.149 --> 00:27:15.740
I would come to that in a minute. In fact,
one particular group, which is S O n, over
00:27:15.740 --> 00:27:22.820
reals is what we are going to understand
in later lectures in much detail. And that
00:27:22.820 --> 00:27:32.909
is going to be very nice application of
group actions. Set a remark, there is this
00:27:32.909 --> 00:27:55.039
remark, and quite often C is not important,
C can be replaced by other fields, say reals,
00:27:55.039 --> 00:28:04.700
or rationals, or what are called finite fields
. Fields, which have finitely many elements,
00:28:04.700 --> 00:28:14.110
or there are some fields, which have a number
theoretic significance, say p-adic fields,
00:28:14.110 --> 00:28:21.700
or local fields, there is so many fields.
So, you can replace them by different fields.
00:28:21.700 --> 00:28:35.200
And particular interest is there, if field
is a like a field numbers, then say in this
00:28:35.200 --> 00:28:41.990
case you have say S O n R that has particular
significance you can geometrically see
00:28:41.990 --> 00:29:08.129
this again, and it occurs as group of rotations
of sphere in R n . So, what is sphere in R
00:29:08.129 --> 00:29:15.239
3, the usually sphere that we imagine. What
is sphere in R 2, circle yeah the usual circle,
00:29:15.239 --> 00:29:23.259
say of sphere or does not matter, you can
say units sphere, which has radius 1, So,
00:29:23.259 --> 00:29:31.570
that is of particular importance . Let me
try to illustrate it for n is equal to 2 case
00:29:31.570 --> 00:29:47.659
that is very easy illustration
So, what is the condition there . S O 2 R
00:29:47.659 --> 00:29:57.580
or another notation is simply call it S O
2, or you just call it S O 2 comma R . So,
00:29:57.580 --> 00:30:10.739
what is the condition. So, condition is S
O 2 is collection of those elements, so M
00:30:10.739 --> 00:30:20.649
in let me write this name G L 2, and I am
having, S O 2 R, G L 2 R, the field is that
00:30:20.649 --> 00:30:37.320
of reals such that M M transpose is 2 by
2 identity matrix, and determinant of M is
00:30:37.320 --> 00:30:51.320
1 .
So, let us try to understand, what are all
00:30:51.320 --> 00:31:00.270
such matrices. So, suppose I take a matrix
a b c d, which is lying here . So, what is
00:31:00.270 --> 00:31:06.279
the property of that property is at a b c
d, when I multiply with its transpose a c
00:31:06.279 --> 00:31:17.149
b d, it is going to have determine, it is
going to have product to be 2 by 2 identity
00:31:17.149 --> 00:31:24.590
matrix. So, what conditions do I get from
this. So, I multiply, I get a square plus
00:31:24.590 --> 00:31:31.720
b square here, and here I get c square plus
d square, therefore both the elements are
00:31:31.720 --> 00:31:38.980
a square plus b square as well as c square
plus b square are 1, cos square theta plus
00:31:38.980 --> 00:31:43.330
sin square theta. Maybe in those terms you
try to understand, and that is how rotations
00:31:43.330 --> 00:31:50.789
come into picture . What are what are other
entries. So, here I have got ac plus bd, which
00:31:50.789 --> 00:31:55.350
has to be 0, and again here I get ac plus
bd .
00:31:55.350 --> 00:32:03.299
So, what are the conditions. Conditions are
a square plus b square is equal to c square
00:32:03.299 --> 00:32:14.830
plus d square is the identity, ac plus bd
is 0, and there is one more quantity, which
00:32:14.830 --> 00:32:25.450
is ad minus bc is a determinant right . Determinant
of this matrix, which is ad minus bc that
00:32:25.450 --> 00:32:37.149
is 1 . So, this is what I get . So, this is
a condition for a b c d. So, one can think
00:32:37.149 --> 00:32:43.490
of a as sin of something, or cos of something,
something cos of theta, so b will be sin theta,
00:32:43.490 --> 00:32:55.950
and let me think of c as a cos alpha, so then
d will be same sin alpha.
00:32:55.950 --> 00:33:01.789
And what about other conditions. So, when
there are ac plus bd, so ac plus bd is cos
00:33:01.789 --> 00:33:16.799
theta cos alpha plus sin theta sin alpha,
and that is 0 . And what about other thing,
00:33:16.799 --> 00:33:38.059
ad minus bc that is cos theta sin alpha
minus, so ac ad minus bc is sin theta cos
00:33:38.059 --> 00:33:53.450
alpha that is also 0 right. So, what is this,
that means cos of theta minus alpha is
00:33:53.450 --> 00:34:06.409
0, . And from here, I get that sin of theta
minus alpha oh sorry this is 1, I just erase
00:34:06.409 --> 00:34:24.230
this, this is 1 the determinant . This is
1, so sin of theta minus alpha is 1 .
00:34:24.230 --> 00:34:41.700
And when I make all these calculations I eventually,
when I saw you all this, what I get is
00:34:41.700 --> 00:35:00.870
that sin alpha is cos theta. And eventually
what I would get is that your matrix a b c
00:35:00.870 --> 00:35:16.120
d is essentially cos theta, and d is again
cos theta something like minus sin theta sin
00:35:16.120 --> 00:35:34.780
theta, and that is rotation matrix . So, these
elements can be thought of as rotation. So,
00:35:34.780 --> 00:35:42.750
this condition, these conditions imply rotation,
and even in the higher dimension, these conditions
00:35:42.750 --> 00:35:46.360
indeed imply rotation. So, this rotation,
which is happening .
00:35:46.360 --> 00:36:01.540
So, in fact, you can think of rotation in
higher dimensions as well . And the remark
00:36:01.540 --> 00:36:17.440
like this rotation matrix, cos theta minus
sin theta sin theta cos theta, this matrix
00:36:17.440 --> 00:36:31.120
can be diagonalized over c complex numbers,
just treat them as complex entries. And what
00:36:31.120 --> 00:36:44.720
you have is e to the power 2 pi i theta e
to the power minus 2 pi i theta 0 0, that
00:36:44.720 --> 00:36:51.040
means, these two matrices are conjugate
to each other, provided you allow conjugation
00:36:51.040 --> 00:36:58.540
by complex numbers, conjugation by entries,
by vertices, which have entries in complex
00:36:58.540 --> 00:37:18.290
numbers .
So, in fact, rotation in R 2, can be thought
00:37:18.290 --> 00:37:42.610
of as complex multiplication . How is that
I can take say cos theta plus i sin theta
00:37:42.610 --> 00:37:58.320
. And then I can think of x plus i y, for
R 3 element, in R 2 x comma y is being identified
00:37:58.320 --> 00:38:07.430
with with x plus i y .
So, when I multiply, what do I get, I get
00:38:07.430 --> 00:38:23.910
for the real entry x cos theta minus y sin
theta plus the complex entry is y cos theta
00:38:23.910 --> 00:38:31.550
plus x sin theta, which is precisely what
this matrix does . So, complex multiplication
00:38:31.550 --> 00:38:38.910
can be thought of as rotation . We are will
talk about higher dimension analog of complex
00:38:38.910 --> 00:38:45.080
multiplication in next lecture, and that is
the case of quarter re-unique multiplication
00:38:45.080 --> 00:38:51.360
quarter are certain objects, and you are going
to learn all that in next lecture. And those
00:38:51.360 --> 00:38:58.480
quarter means are helpful in the in the
understanding rotation in not in R 2, but
00:38:58.480 --> 00:39:03.470
R 3 .
So, what we have seen today that abstract
00:39:03.470 --> 00:39:14.010
elements in group can be can be thought of
as matrices as permutations, they are useful
00:39:14.010 --> 00:39:19.840
in other things like rotations, and all that
into the connection with complex multiplication
00:39:19.840 --> 00:39:32.970
. Next time, we are going to talk about
what is called a representation in the formal
00:39:32.970 --> 00:39:45.920
way, which is an attempt to make every
element as a matrix.
00:39:45.920 --> 00:39:50.250
And there are certain applications, certain
advantages of representations, we are going
00:39:50.250 --> 00:39:56.040
to talk about mathematical representations
next time. But, as of now I am sure, you have
00:39:56.040 --> 00:40:02.350
understood that groups can be represented
in various forms, in forms of puzzles, in
00:40:02.350 --> 00:40:07.740
forms of some permutation games, in form of
some symmetric considerations symmetric considerations,
00:40:07.740 --> 00:40:14.170
I will show you all those symmetric objects
. So, to know more about representations,
00:40:14.170 --> 00:40:18.440
watch for the next video.
Thank you.