CM::Group::Sym - An implementation of the finite symmetric group S_n
CM::Group::Sym is an implementation of the finite Symmetric Group S_n
use CM::Group::Sym;
my $G1 = CM::Group::Sym->new({$n=>3});
my $G2 = CM::Group::Sym->new({$n=>4});
$G1->compute();
$G2->compute();
This way you will generate S_3 with all it's 6 elements which are permutations. Say you want to print the operation table(Cayley table).
print $G1
6 5 4 3 2 1
3 4 5 6 1 2
2 1 6 5 4 3
5 6 1 2 3 4
4 3 2 1 6 5
1 2 3 4 5 6
or the table of S_4 with 24 elements
print $G2
24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
23 24 20 16 22 18 19 15 21 17 13 14 11 12 8 4 10 6 7 3 9 5 1 2
22 18 19 15 23 24 20 16 11 12 8 4 21 17 13 14 9 5 1 2 10 6 7 3
21 17 13 14 9 5 1 2 10 6 7 3 22 18 19 15 23 24 20 16 11 12 8 4
20 19 18 17 24 23 22 21 12 11 10 9 16 15 14 13 4 3 2 1 8 7 6 5
19 20 24 12 18 22 23 11 17 21 9 10 15 16 4 8 14 2 3 7 13 1 5 6
18 22 23 11 19 20 24 12 15 16 4 8 17 21 9 10 13 1 5 6 14 2 3 7
17 21 9 10 13 1 5 6 14 2 3 7 18 22 23 11 19 20 24 12 15 16 4 8
16 15 14 13 4 3 2 1 8 7 6 5 20 19 18 17 24 23 22 21 12 11 10 9
15 16 4 8 14 2 3 7 13 1 5 6 19 20 24 12 18 22 23 11 17 21 9 10
14 2 3 7 15 16 4 8 19 20 24 12 13 1 5 6 17 21 9 10 18 22 23 11
13 1 5 6 17 21 9 10 18 22 23 11 14 2 3 7 15 16 4 8 19 20 24 12
12 11 10 9 8 7 6 5 4 3 2 1 24 23 22 21 20 19 18 17 16 15 14 13
11 12 8 4 10 6 7 3 9 5 1 2 23 24 20 16 22 18 19 15 21 17 13 14
10 6 7 3 11 12 8 4 23 24 20 16 9 5 1 2 21 17 13 14 22 18 19 15
9 5 1 2 21 17 13 14 22 18 19 15 10 6 7 3 11 12 8 4 23 24 20 16
8 7 6 5 12 11 10 9 24 23 22 21 4 3 2 1 16 15 14 13 20 19 18 17
7 8 12 24 6 10 11 23 5 9 21 22 3 4 16 20 2 14 15 19 1 13 17 18
6 10 11 23 7 8 12 24 3 4 16 20 5 9 21 22 1 13 17 18 2 14 15 19
5 9 21 22 1 13 17 18 2 14 15 19 6 10 11 23 7 8 12 24 3 4 16 20
4 3 2 1 16 15 14 13 20 19 18 17 8 7 6 5 12 11 10 9 24 23 22 21
3 4 16 20 2 14 15 19 1 13 17 18 7 8 12 24 6 10 11 23 5 9 21 22
2 14 15 19 3 4 16 20 7 8 12 24 1 13 17 18 5 9 21 22 6 10 11 23
1 13 17 18 5 9 21 22 6 10 11 23 2 14 15 19 3 4 16 20 7 8 12 24
Note that those are only labels for the elements as printing the whole permutations would render the table useless since they wouldn't fit. You can find that for S_5 the table would not fit on the screen (or maybe it would if you had a big enough screen, or a small enough font).
So if you want to see the meaning of the numbers(the permutations behind them) you can use str_perm()
print $G1->str_perm;
1 -> 3 2 1
2 -> 2 3 1
3 -> 2 1 3
4 -> 3 1 2
5 -> 1 3 2
6 -> 1 2 3
find the conjugacy classes using the definition of conjugates.
finds the conjugacy classes of a group using cycle structure. for example, the conjugacy classes of S_4 correspond to partitions of the number 4:
(x)(x)(x)(x)
(xx)(x)(x)
(xx)(xx)
(xxx)(x)
(xxxxx)
so S_4 has 5 conjugacy classes.
Computes the operation table.
This method will draw a diagram of the group to png to the given $path. You can read the graph as follows. An edge from X to Y with a label Z on it that means X * Z = Y where X,Y,Z are labels of permutations.
return the identity permutation for this group
this method returns the centralizer of an element in the group. (note that the centralizer can be different in some particular subgroup)
returns the center of the group
computes the normal subgroups of a group of permutations
given a set of generators @S , Dimino's algorithm(see [2] for details) enumerates all elements of the subgroup <@S> generated by the set @S
The implementation is using some theorems(such as Lagrange's theorem, the class equation, Cauchy's theorem) as tests. This way of using theorems offers some sense of security that what was implemented is indeed correct. Check the tests attached to this distribution for more details.
Groups are not as abstract as they seem. Some permutation groups can relate to geometric objects such as a cube, a tetrahedron or a dodecahedron(actually, all platonic solids have associated rotation group, which is a group of isometries that fixes a particular regular polyhedron).
There is a strong relation between S_4 and the group of rotations of a cube. In a similar way there is an isomorphism between A_5(the even permutations of 5 objects) and the group of rotations of the dodecahedron and also between A_4 and the group of rotations of a tetrahedron and also between the groups of rotations of an icosahedron and A_5.
Patches/suggestions are always welcome.
[1] Joseph Rotman - An Introduction to the Theory of Groups
[2] Gregory Butler - Fundamental Algorithms for Permutation Groups (Lecture Notes in Computer Science)
Stefan Petrea, <stefan.petrea at gmail.com>