25 April 2014 ~ 0 Comments

This week: The isomorphism problem once again

Pursuing my study of presentations of groups, I worked mainly this week on GAP adding some functions to my package. These functions are:

  •  DanDisplayTree( Tree, Gens, TypeDisplay )*  
  • DanExtractTreeFromTable( CT, Generators, Sec )*

and I used them in the following GAP program:

disp := “C:\\Users\\papa\\Desktop\\MesProgrammes\\Tests\\toto2.txt” ;
OutputLogTo( disp ) ;

ShowPackageVariables(“Daniel”) ;

G := SmallGroup(16,2) ;
Print(DanDisplayGroup(G)) ;
M := MultiplicationTable(G) ;

Gens := [2,3] ;
Arbre := DanExtractTreeFromTable(M,Gens,30) ;
DanDisplayTree(Arbre,Gens,0) ;

Gens := [3,2] ;
Arbre := DanExtractTreeFromTable(M,Gens,30) ;
DanDisplayTree(Arbre,Gens,0) ;

This program gives the following results:

—————————————————————-
Loading  Daniel1.0.0
by Daniel Dupard (http://www.maths77.fr/daniel/)
For help, type: ?Daniel package
—————————————————————-
new global functions:
  DanAdditionTable( CT )*
  DanAdditionTableWithoutModulo( CT )*
  DanAutomorphisms( arg )*
  DanCenter( G )*
  DanCyclicSubgroups( G )*
  DanDisplayGroup( arg )*
  DanDisplaySubgroups( G )*
  DanDisplayTable( CT )*
  DanDisplayTree( Tree, Gens, TypeDisplay )*
  DanExtractTreeFromTable( CT, Generators, Sec )*
  DanExtractTrianglesFromGroup( arg )*
  DanInnerAutomorphisms( arg )*
  DanInverseElements( arg )*
  DanPermuteTable( CT, P )*

other new globals (not write protected):
  DoIContinueCreatingTheTree( Found )*
  MatrixValue( v, OrderGRP )*

C4 x C4
  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
  2  4  6  7  8  9  1 11 12 13 14  3 15  5 16 10
  3  6  5  9 10  8 12 13 11  1 15 14  2 16  4  7
  4  7  9  1 11 12  2 14  3 15  5  6 16  8 10 13
  5  8 10 11  1 13 14  2 15  3  4 16  6  7  9 12
  6  9  8 12 13 11  3 15 14  2 16  5  4 10  7  1
  7  1 12  2 14  3  4  5  6 16  8  9 10 11 13 15
  8 11 13 14  2 15  5  4 16  6  7 10  9  1 12  3
  9 12 11  3 15 14  6 16  5  4 10  8  7 13  1  2
 10 13  1 15  3  2 16  6  4  5  9  7  8 12 11 14
 11 14 15  5  4 16  8  7 10  9  1 13 12  2  3  6
 12  3 14  6 16  5  9 10  8  7 13 11  1 15  2  4
 13 15  2 16  6  4 10  9  7  8 12  1 11  3 14  5
 14  5 16  8  7 10 11  1 13 12  2 15  3  4  6  9
 15 16  4 10  9  7 13 12  1 11  3  2 14  6  5  8
 16 10  7 13 12  1 15  3  2 14  6  4  5  9  8 11

function DanDisplayTree
a = 2  aa = 4  aaa = 7  aaaa = 1 
a = 2  aa = 4  aaa = 7  aaab = 12  aaaba = 3 
a = 2  aa = 4  aaa = 7  aaab = 12  aaabb = 14  aaabba = 5 
a = 2  aa = 4  aaa = 7  aaab = 12  aaabb = 14  aaabbb = 16  aaabbba = 10 
a = 2  aa = 4  aaa = 7  aaab = 12  aaabb = 14  aaabbb = 16  aaabbbb = 7 
a = 2  aa = 4  aab = 9  aaba = 12 
a = 2  aa = 4  aab = 9  aabb = 11  aabba = 14 
a = 2  aa = 4  aab = 9  aabb = 11  aabbb = 15  aabbba = 16 
a = 2  aa = 4  aab = 9  aabb = 11  aabbb = 15  aabbbb = 4 
a = 2  ab = 6  aba = 9 
a = 2  ab = 6  abb = 8  abba = 11 
a = 2  ab = 6  abb = 8  abbb = 13  abbba = 15 
a = 2  ab = 6  abb = 8  abbb = 13  abbbb = 2 
b = 3  ba = 6 
b = 3  bb = 5  bba = 8 
b = 3  bb = 5  bbb = 10  bbba = 13 
b = 3  bb = 5  bbb = 10  bbbb = 1 
function DanDisplayTree
a = 3  aa = 5  aaa = 10  aaaa = 1 
a = 3  aa = 5  aaa = 10  aaab = 13  aaaba = 2 
a = 3  aa = 5  aaa = 10  aaab = 13  aaabb = 15  aaabba = 4 
a = 3  aa = 5  aaa = 10  aaab = 13  aaabb = 15  aaabbb = 16  aaabbba = 7 
a = 3  aa = 5  aaa = 10  aaab = 13  aaabb = 15  aaabbb = 16  aaabbbb = 10 
a = 3  aa = 5  aab = 8  aaba = 13 
a = 3  aa = 5  aab = 8  aabb = 11  aabba = 15 
a = 3  aa = 5  aab = 8  aabb = 11  aabbb = 14  aabbba = 16 
a = 3  aa = 5  aab = 8  aabb = 11  aabbb = 14  aabbbb = 5 
a = 3  ab = 6  aba = 8 
a = 3  ab = 6  abb = 9  abba = 11 
a = 3  ab = 6  abb = 9  abbb = 12  abbba = 14 
a = 3  ab = 6  abb = 9  abbb = 12  abbbb = 3 
b = 2  ba = 6 
b = 2  bb = 4  bba = 9 
b = 2  bb = 4  bbb = 7  bbba = 12 
b = 2  bb = 4  bbb = 7  bbbb = 1 
gap>
I plan to work on a function which compares trees and then on a clever choice of generators for a Cayley Table. I have several ideas to reduce the number of generators combinations to study before being able to demonstrate that 2 tables are isomorphic or not till a maximum order which I hope will be more than 128.

 

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