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Language: en
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So, the last lecture, we are talking about
quotienting of groups. And I said that when
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you have a group, which is its normal subgroup
of bigger group G, so H is normal subgroup
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of G, then
quotienting makes sense. Today, I will make
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it bit more precise. So, to start with let
us have a group, and any subgroup ; A subgroup
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meaning, a subset of G, which is group in
itself under the same operation, which is
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defined on G .
So, what you can do is you can consider, what
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are called cosets, so that we take any element
in G, and you consider say a H. What is a
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H by definition, it is a collection of all
the elements of the type a H such that h is
00:02:06.190 --> 00:02:21.340
in H. So, this is a subset of G . So, this
is so called left coset, a is coming to the
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left. We are having group operation, we
are multiplying a from the left side, so that
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is left coset . I know what you do, you consider
all left cosets .
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So, here I am considering a H such that a
is in G. So, for two different a in G, so
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say a 1, and a 2 different . You may actually
have same coset a 1 H, a 2 H, it is quite
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possible . So, you have this collection of
cosets. So, this is set of cosets of G by
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H, so these are cosets. And one has to
observe
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that if I take two element, say a, and b,
then either a H is same as b H, or subset
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a H does not intersect subset b H. So, this
intersection is taking place in G, intersection
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is happening in G, so that means G can be
written as union and disjoint union, disjoint
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union of these courses .
Let us come back to the set of cosets, I will
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denote it by G mod H it is notation . And
on this G mod H, I try to define group operation.
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How do I do, I just trying to say that if
I take one element here, this is a coset,
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another element, which is here it is a coset,
I I define it a b H. Then in order to say
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that all this makes sense . This operation
should be well defined, so that is all is
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a question, is it well defined, because the
choice of a, could be something else, choice
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of b, could be something else . So, choice
of a could be something, so that a H is same
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as a dash H, some other choice right.
Similarly, we could have some b dash H, such
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that b dash H is same as b h, but b is different
from b, b dash is different from b . Then
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because choice is involved the question of
well definedness is certainly there. And it
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is a proposition that if H is normal, H normal
subgroup. I hope you define what normal subgroup
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is normal subgroup is the one, for which for
every choice of G in G, and for every choice
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of H in H, the conjugation G and G inverse
belongs to H . So, if H is normal, then this
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operation that I have defined, it is well
defined. And in that situation, if you consider
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set this along with the operation that has
been obtained in
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this fashion, and it is actually a group.
So, star, this star operations actually group
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operation, so that is about quotienting . Quotienting
has many applications .
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Let us see some quick examples of quotienting
. I have G is equal to Z, integers H is equal
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to 2 Z, even integers. Then G mod H is just
group of two elements . Group of two elements,
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just write as say 0 1 . Why am I writing it
as 0 1, here 1 plus 1 is 0, why am I writing
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like this . Because what are the cosets, cosets
would be cosets corresponding to identity,
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so this is 2 Z, and then 1 plus 2 Z, these
are the cosets . And this called coset corresponds
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to 0, this coset corresponds to 1 is a trivial
example of quotienting .
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Ah Now, I will discuss some more concepts,
remember what is the purpose, purpose to understand
00:09:04.190 --> 00:09:10.540
Rubik's group, and you solve Rubik's group.
Some some quotienting will be required there,
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and some understanding of generators and relations
expressing a group, in terms of generators
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and relations will also be required there
ok .
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So, let us see few more things homomorphisms
. So, what is the meaning homomorphism of
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the same type right. So, suppose I have a
group G, I have another group H, it is a map
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between these function from G to H, function
f I called homomorphism. If whenever I have
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two element say a and b here in G, they actually
map to the a into b actually maps to f a f
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b .
What is this, this is the group law in H.
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What is this, this is group law in G . And
so so what what is being said that is f of
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a b is same as f of a f of b. Image of the
product is product of images. By product what
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do what do we mean, we mean the group operation,
whichever is there in the relevant group H
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and G . And one of the consequences of this
is that identity of G has to go to identity
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of H. Why, because you put a and b both to
be identity, identity of G, and then you see
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what happens ok . So, these maps are called
homomorphisms.
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What are the examples .
We have seen some of them, you remember
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signature map . What was that from symmetry
group, symmetry group on n letters to say
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the group 0 1 . 0 1 group is what, we have
seen it is just Z mod 2 Z odd and even odd
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plus odd is even .
So, permutation sigma goes to signature of
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sigma. I hope we recall from previous lectures
that for a permutation we can assign, notion
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of signature. What is that every permutation
is a product of transpose, transpose meaning
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two elements getting swapped . So, if there
are odd number of transpose we say the signature
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is 1 even number of transpose, which are involved
in expressing sigma, then you have signature
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to be 0 .
So, if you have sigma n theta, which are two
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permutations, the signature of them is
actually signature of sigma plus signature
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of theta, and this signature is this addition
is actually happening in Z mod 2 Z, so that
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is one example of homomorphisms. So, you
can see this, this is precisely the defining
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property of homomorphisms.
Another example I can take again quite simple
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one . I take integers, and then I take cyclic
group . What is a cyclic group this one, by
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definition this is I say in terms of generator
and relations, it is generated by an element
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a such that a to the power n is 1 ok . So,
what are the elements here, elements are 1
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identity that is a, a square, and so on up
to a to the power n minus 1. So, these n elements
00:14:30.710 --> 00:14:38.470
are there, and product is formal . So, a to
the power i a to the power j, it is simply
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a to the power i plus j, and this i plus j
is happening mod n yeah .
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So, I just take say and then you are
going to say r I map it to r mod n, so here
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I would say a to the power r mod n . So, whatever
is the remainder after dividing r by n, so
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that remainder is going to be from 0 to n
minus 1, and that as a power of a and that
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is a homomorphism. So, it is easy to check
that let me call this map f, f is a homomorphism
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.
Some more examples of homomorphism, I will
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mention
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that is interesting one . You can see the
real numbers with 0 removed, and then you
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take the ones, which are positive, or in other
words what you are taking are simply positive
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real numbers . So, positive real numbers together
with usually multiplication form a group .
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And then you take real numbers together with
addition . So, from positive real numbers,
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so I would say R strictly positive. So, this
is R strictly positive so, R strictly positive.
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Together with multiplication, just to indicate
verification, and then R, the addition ; I
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define this map, when you have known this
map for quite long time log to the base let
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me just say 10 for simplicity . So, a positive
real goes to log of x, and the base is 10
00:18:02.629 --> 00:18:08.500
.
You remember from your school days, at log
00:18:08.500 --> 00:18:27.039
of x y is same as log of x plus log of y that
creation we have seen what was that, it just
00:18:27.039 --> 00:18:40.830
said that this homomorphism condition, which
is satisfied . So, this map log say to the
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base 10 is a homomorphism, does not matter
what base is log to any base is a homomorphism
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. Here it is multiplication and here it is
addition, so that is very interesting example
00:19:01.179 --> 00:19:02.429
of homomorphism.
And in fact, this homomorphism is invertible.
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How is that, given any real number, does not
matter positive or negative or 0, any given
00:19:15.379 --> 00:19:24.950
any real number, I can raise it to the power
of y . What we called if you remember during
00:19:24.950 --> 00:19:41.789
this school days, antilog, or it is also call
it exponent exponential, exponential with
00:19:41.789 --> 00:19:51.019
respect to 10. So, for log, we have antilog
map, and it combines well, it is a identity
00:19:51.019 --> 00:20:10.330
. So, such homomorphisms, which are invertible,
they are called isomorphism. So, invertible
00:20:10.330 --> 00:20:19.989
homomorphisms are called isomorphisms definition
yeah.
00:20:19.989 --> 00:20:47.659
Let us see some more examples of homomorphisms
. You consider this set, what is that n by
00:20:47.659 --> 00:21:04.019
n matrices with real entries with entries
in . Is
00:21:04.019 --> 00:21:10.489
this a group, when you have to tell the operation
under addition, it is a group under multiplication
00:21:10.489 --> 00:21:17.980
it is not a group, because there are so many
matrices, which are not invertible under multiplication
00:21:17.980 --> 00:21:28.639
.
So, what I do, I consider those matrices with
00:21:28.639 --> 00:22:04.850
entries are with entries in R, which have
a multiplicative inverse . There that forms
00:22:04.850 --> 00:22:29.580
a group that forms a group under matrix multiplication
. What is the identity of this group, like
00:22:29.580 --> 00:22:53.169
that identity matrix; the matrix, which has
all diagonal entries as 1, and all other entries
00:22:53.169 --> 00:23:02.799
as 0 .
So, from this matrix to real numbers with
00:23:02.799 --> 00:23:11.669
multiplication So, when I take multiplication,
I have to remove 0. So, these are non-zero
00:23:11.669 --> 00:23:20.880
real numbers . I have to remove 0 with multiplication
. Here is a map that we have seen in the school,
00:23:20.880 --> 00:23:40.909
which is a homomorphism, can you guess it,
given any matrix I am going to associate it
00:23:40.909 --> 00:23:48.289
to scalar determinant . You take a matrix
A, you associate to it the determinant of
00:23:48.289 --> 00:24:05.929
this matrix . And then as you recalled, determinant
of A into B is same as determinant of A times,
00:24:05.929 --> 00:24:13.960
determinant of B . What is it, homomorphism,
so determinant is a homomorphism.
00:24:13.960 --> 00:24:31.080
Is this may have an isomorphism, that means
does there exist a map. In the reverse direction,
00:24:31.080 --> 00:24:41.239
to which if I compose determinant I get identity,
no because there could be two different matrices
00:24:41.239 --> 00:24:54.730
with the same determinant ; So, it is not
an isomorphism, because two different matrices
00:24:54.730 --> 00:25:04.739
of same determinate may exist do exist, so
that is not an isomorphism .
00:25:04.739 --> 00:25:24.419
Let us see one more example . You
remember the direct product of groups, what
00:25:24.419 --> 00:25:46.610
is it, it is 0 0, 1 0, 0 1, 1 1, and then
you consider V 4, which I wrote as 1 f x,
00:25:46.610 --> 00:26:10.549
f y, r pi. When I am writing like this, I
want to emphasize that this is group of
00:26:10.549 --> 00:26:14.960
symmetries of a rectangle . So, one can actually
give an isomorphism from here to here, from
00:26:14.960 --> 00:26:26.820
Z 2 cross Z 2 to V 4 . So, of course, is a
isomorphism 0 0 has to go to 1. What about
00:26:26.820 --> 00:26:34.139
the other elements, there are many isomorphisms
from Z 2 cross Z 2 to V 4, maybe that is an
00:26:34.139 --> 00:26:54.080
assignment question. So, how many isomorphisms
are there ok . So, after this discussion on
00:26:54.080 --> 00:26:59.610
isomorphisms, and the quotienting of groups
homomorphisms.
00:26:59.610 --> 00:27:13.580
Let me quickly discuss one more concept,
which we are going to see in the Rubik's
00:27:13.580 --> 00:27:38.950
cube case, Cayley graph of a group . So, given
a group, and a subset of it, which is generating
00:27:38.950 --> 00:27:48.529
set, I can think of it is graph, the Cayley
graph of the group . So, what kind of graph
00:27:48.529 --> 00:27:58.299
is this . So, first of all what a graph is,
graph is any pictorial representation, where
00:27:58.299 --> 00:28:06.090
there are several nodes, there are several
vertices, and they are connected to each other
00:28:06.090 --> 00:28:21.379
by a certain rule something . So, there are
nodes, these nodes we calls, so call them
00:28:21.379 --> 00:28:37.499
vertices, and then these are edges .
So, in a Cayley graph, what are the nodes.
00:28:37.499 --> 00:28:44.259
Nodes are actually all the elements, so I
am assuming group is finite. So, all the elements,
00:28:44.259 --> 00:29:08.669
which are there in the group, I take them
as nodes . Nodes are labeled by G. And what
00:29:08.669 --> 00:29:18.649
about edges. So, to define edges, I have to
specify that I am writing, I am constructing
00:29:18.649 --> 00:29:32.309
the Cayley graph with respect to what set
of generators, set of generators .
00:29:32.309 --> 00:29:49.450
So, what do what we do, suppose these are
nodes, these are elements of G. Then here
00:29:49.450 --> 00:30:04.399
is a element a, here is an element b . I would
connect a to b, so edge edges are. So, we
00:30:04.399 --> 00:30:31.379
drew an edge between a and b, if either a
inverse b belongs to the generating set, or
00:30:31.379 --> 00:30:40.110
b inverse a belongs to the generating set.
So, if one of them belongs to the generating
00:30:40.110 --> 00:30:49.299
set, then you connect the edges.
Let us see with the example. And your example
00:30:49.299 --> 00:31:15.289
is a favorite example of say D 4 . So, I remember,
I hope you also remember that that for D 4,
00:31:15.289 --> 00:31:26.793
one of the generating sets is r comma f. This
is rotation by 90 degree, and this was a about
00:31:26.793 --> 00:31:36.129
y axis, this is rotation by 90 degree . So,
D 4 is generated by this r and f .
00:31:36.129 --> 00:31:52.860
And how to draw the Cayley graph; so, what
are the elements in D, I have identity, I
00:31:52.860 --> 00:32:06.749
have r, I have r square, I have r cube . So,
one will get connected to r, because r times
00:32:06.749 --> 00:32:16.149
1 inverse. What are the conditions, the conditions
were a inverse b belongs to s, or b inverse
00:32:16.149 --> 00:32:27.570
a belongs to s, these were the conditions.
So, here 1 inverse r belongs to s. So, I am
00:32:27.570 --> 00:32:38.220
connecting similarly, r belongs to s, r belongs
to s, and here also I can make these connections.
00:32:38.220 --> 00:32:51.779
So, these are four nodes right. And then one
is also connected to f.
00:32:51.779 --> 00:33:01.590
And then what are other elements, f is there,
f r is there as a node, f r square is there
00:33:01.590 --> 00:33:12.529
as a node f r cube . So, these are nodes
. And I am connecting f with f r, because
00:33:12.529 --> 00:33:22.820
f inverse f r like belongs to s that is r.
And similarly, I connect this, I connect this,
00:33:22.820 --> 00:33:31.730
I connect this, I also connect r and f r,
and I also connect, so I am connect connecting
00:33:31.730 --> 00:33:37.099
r and f r, because f r r inverse, they belongs
to the set s .
00:33:37.099 --> 00:33:47.450
And similarly, I have to connect r square
with f r square, and r cube with f r cube.
00:33:47.450 --> 00:33:59.859
You realize what we have obtained, it is actually
cube. So, the Cayley graph of D 4 is a cube,
00:33:59.859 --> 00:34:15.890
where if you want I can label like 1, r, r
square, r cube, and here f, f r, f r square,
00:34:15.890 --> 00:34:24.640
f r cube. So, this is a Cayley graph of
D 4. So, in fact, whenever there is permutation,
00:34:24.640 --> 00:34:27.820
whenever there is a action, which is happening
.
00:34:27.820 --> 00:34:35.040
When one try to identify what are the basic
moves, those basic moves you can think of
00:34:35.040 --> 00:34:40.000
as generating sets. And then for the whole
game, for the whole sequence of permutations,
00:34:40.000 --> 00:34:49.300
one can create a Cayley graph, and traversing
Cayley graph is a fun . There are many
00:34:49.300 --> 00:34:59.270
other situations, where drawing Cayley graphs
is important. One subject area of mathematics
00:34:59.270 --> 00:35:12.060
is what is called growth of groups, which
concerns how these Cayley graphs grow with
00:35:12.060 --> 00:35:17.300
respect to a set of generators, and what is
the effect of choice of set of generators
00:35:17.300 --> 00:35:22.530
on this growth .
So, if you want, I will give name of the person
00:35:22.530 --> 00:35:33.600
Michael Gromov, who is one of the pioneers
of this area. But, we are not going to study,
00:35:33.600 --> 00:35:40.730
and growth of groups in this course, we have
just interested in various applications
00:35:40.730 --> 00:35:47.560
of groups. And in particular in some of
the situations graph may come, it would be
00:35:47.560 --> 00:35:53.359
great fun for you to draw a graph of say 15
puzzle, or to draw a graph of Rubik's cube
00:35:53.359 --> 00:36:02.040
. Can you imagine, how large will the graph
of Rubik's cube be, any idea? 1 million, 1
00:36:02.040 --> 00:36:10.930
billion, may be more, you will see that we
are going to play with Rubik's cube using
00:36:10.930 --> 00:36:21.170
a software, which is called GAP - Groups Algorithms
and Programming. You have to watch keep enjoy.
00:36:21.170 --> 00:36:22.600
Thank you.