Isomorphism problem is a hard problem which is known to be decidable for most groups since it is decidable for hyperbolic groups (see arXiv:1002.2590 ”The isomorphism problem for all hyperbolic groups” from François Dahmani, Vincent Guirardel). However we know large classes of groups for which this problem is undecidable like

- free by free groups (Miller 71)
- free abelian by free groups (Zimmermann 85)
- solvable groups of derived length 3 (Baumslag-Gildenhuys-Strebe 85)

Even if the problem of isomorphism is decidable for finite groups (finite groups are hyperbolic), there are some interesting things to learn on this subject. This work has given me some insights on another subject I love which is: “why some theories are complete and some are not ?”

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Today is a great day. I work hard since several months (neally a year) on the isomorphism problem in groups (google it, if you don’t know what is it).

Having worked neally all the week end on an idea that surged last week, I can say that for the first time, If you give me 2 Cayley Tables of finite groups, I can say if these Cayley Tables are isomorphic or not by computing special normalized presentations. It probably does not work in all cases but it works for a lot of them.

I know some cases for which what I have done, does not work, but I know how to improve quite easily these cases. There are probably some couples of Cayley Tables for which I can’t say if they are isomorphic or not, but for the moment I don’t see them.

Actually I feel good, very good. It won’t last (as usual) because I will pretty soon find another wall but for the moment I’m just peaceful.

I should definitely study in detail power commutator presentations (I think Sylow was their discoverer) which are used by Eamon O’Brien to compute the groups of orders 256 and 512 and PolyCyclic Generating Sequences which are used by Bettina Eick to compute automorphisms of p-groups. Knowing these specific presentations, I could compare them with my own presentation and study their respective properties.

]]>- 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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With Internet, human kind created a new library of babel. Actually this library is still small but growing rapidly. In his novel (La biblioteca de Babel) Borges spoke about a library containing all books that could be written with an alphabet of 22 letters, the period, the comma and the space. Actually such a library already exists and it’s not internet. Moreover there is not one library of this type, but a lot.

No idea ?

I am speaking about irrational numbers. In their infinity of decimals one can find all the books which has been written and who will be written in the future. Each irrational contains in its list of decimals the torah, the bible and the coran (to respect the chronological order) somewhere. Moreover what I am writing actually is somewhere else, not in one irrational but in all irrationals. Just think about it.

Whatever I added 3 links to myblog. They illustrate perfectly the richness of Internet nowadays.

The first one added “Magmas, Loops and Monoids” is a wonderful introduction to the most simple algebraic structures. It was written by Konrad Voelkel, a PHD student.

The second link added is an aggregator of mathematical blogs, to keep in touch with what is written in the mathematical world.

The third link added is arxiv. Just have a look if you don’t know it

]]>In fact, after having found something possibly interesting (object, property or conjectures), I immediately search for articles or books on the same subject or on subjects which are linked one way or another with the idea I have. This is the most fundamental lesson you learn when you do a PHD: “study what has been done before you”. Know who has written what, and read it.

Sometimes, the idea I have, has been explored nicely and completely by others. Sometimes I do not find any article, so I explore the concept the more I can, before discovering that the subject has been long studied and that I didn’t know the terms (the vocabulary) to use to find the articles on this subject. Sometimes, I’m not able to explore the idea I have. It’s too complicated (at least for me). And sometimes it’s an original idea. Original does not mean great. Most of the time the idea I have, is not a great one. However when I understand it, I still enjoy having learned something, even it is unsignificant, even if it’s a tiny step, because it always indicates me where could be some other steps.

For example, triangles in groups are not a great idea, however I know when they occur (I demonstrated it) and most importantly when they do not occur. In this cases, can I find some other special symmetries ?

Symmetries are fascinating, they give a sort of order in disordered world. The best book on this subject for everyone and not for a specialist of this subject is “Symmetry: A Journey into the Patterns of Nature” from Marcus Du Sautoy. You don’t need to be a mathematician to understand this book. It is well written and can be really read like a novel.

Groups (as a mathematical object) are built upon symmetry. That’s why they are not so numerous. In fact they are very scarce. There are far more magmas than groups at a given order.

]]>- Mathematics,
- Teaching,
- Informations,
- Books.

In this edition, we can find an article about the demonstration of the ternary form of “The Goldbach conjecture” by Harald A. Helfgott. Last year, I studied this subject for a talk. This article is a very nice introduction to this subject.

Another article concerns the collaborative project polymaths launched by Tim Gowers, Michael Nielsen, Terence Tao and supported by many mathematicians around the world.

This project (polymath8) was about Yitang Zhang’s demonstration that established that bounded gaps between primes occur infinitely often. This project was an incredible success and this inititative is a wonderful idea. You can find more info on this on the wiki below:

In the Teaching part of this journal, we find mostly articles about how to teach mathematics. These questions always interest me even if I’m not a teacher and most of what I know about mathematics have been learned in books and articles. For most of us (if not all), the foundations come from school and the content of what is learned at school is a crucial point for the next generations (as it was for me). From what you learn at school you can then study by yourself, but the more the foundations are solid, the more you can work easily.

Actually in France, we study less mathematics at school till the age of 18 than 30 years ago. It has some consequences on teaching mathematics in University.

Moreover there is an article about Marc Yor (died in January 2014) who was a very kind person and a brilliant mathematician. He misses me.

]]>It’s rather easy to create a magma since the only thing you have to do is to choose a set (for example the numbers from 0 to 9) and to create an addition table where the result of the addition of 2 elements is randomized between the elements of the set (for example 1+1=3). Using this method the probability to create a magma is neally 100% when the set S contains a sufficient number of elements.

What magmas did I worked with ?

I just used the following operation. Let S be the set of natural integers

if a > 0 then a*b = b + (a-1) f(b)

if a = 0 then a*0 = 0

where f(b) can be choosen at will.

if for each b in intergers, f(b) = b then we have the multiplication everyone studied at primary school. if f(b) <> b for some b then we start to have special properties.

This operation is a kind of generalization of the well known multiplication.

Why is it useful to work with magmas ?

In my case I use magmas to work on some triangles in groups. I discovered some times ago that there were triangles in groups. I conjectured some ideas and then worked on demonstrations of these conjectures. For one of my first conjecture (in the case of cyclic groups), I used presentation of groups. For the group of quaternions, I used presentations too.

But to understand why there were some triangles in a group, I have to understand which special property of a group creates triangles. The good way to do this is to use algebraic structure with no special property and to add little by little some new properties. This is what I’m working on.

]]>So I created a GAP program to give me the presentations of groups from 1 to the order 128.

disp := “C:\\Users\\papa\\Desktop\\MesProgrammes\\Tests\\Presentations.txt” ;

OutputLogTo( disp ) ;

str := “” ;

output :=OutputTextString( str, true);

for i in [1..128] do

n := NumberSmallGroups(i) ;

for j in [1..n] do

SizeScreen([10000,]) ;

G := SmallGroup(i,j) ;

iso := IsomorphismFpGroup(G) ;

fp := Image(iso) ;

PrintTo(output,”GAP(“,i,”,”,j,”) = < “,GeneratorsOfGroup(G),” | “,RelatorsOfFpGroup(fp),” >\n”) ;

od ;

PrintTo(output,”\n”) ;

od ;

CloseStream(output);

Print(str) ;

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