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 Combinatorial Groups

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Dr. Post
Pythagoras
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PostSubject: Re: Combinatorial Groups   Wed Jan 05, 2011 12:18 pm

It is not about poetry here.

It is about Consumer Protection. The definitions sellers should specify the ground group of their products, for the case that someone want to use them as labels.

Of course, the ground group of a ground group is right the ground group so let's go on.
-----------

Here is a good news for you Nick : Rubik' Cube contains two wreath products : S8(Z3)×S12(Z2). (Wikipedia)

Meanwhile, it is a factor of the above by Z2×S3. Par Herr Holder, I think the factorization should be considered next operation with ground groups.
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nick
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PostSubject: Re: Combinatorial Groups   Wed Jan 05, 2011 9:11 pm

I am not convinced doc, about the fifth operation. That factorization kills everything...

One can loose a lot :
for degree 5

The field F5 and the Diamond

for degree 6 (at least)

a nice degenerated platonic solid Kd = (XX(E2(XX)))/Z2 which is a Platonic Solid,
1/2+1/2+1/2 = 6/4 : two faces, two edges, two vertices...

==============
Let's take a simpler Rubik cube, just 3 corners on the same face. The species is a "factor" of

Ens3(Cyc3)

Just take out those three corners and put them on the table, buildinga 3-set of 3-cycles.
The orbit has cardinality 162= 6×27 (the order of Stab)
When mounted, the orbit has only 27 cardinality. because one cannot rotate a corner or swap two corners.

This "factorization" could mean to "lock" a cycle or something, analogue to disintegration, when "stick" something. Par Rubik !, if there is a "factorization", it should be associated to the combinatorial composition.

==============
And here are already listed Rubiks ;

Firstly let's introduce
- odd species and even species, depending on the parities in the Stab, which has either half even permutations or only even permutations.
- the taquinization, meaning that in one odd species we consider only half of relabellings, the even ones.

Then :

Ens --- taquinization ----> Alt

Yves Chiricota, page 35, lines 30 and 29, mean :

Ens3( Ens2) --- taquinization ----> Rubik on 3 edges

=======
it seams that the juicy species remain after the elimination of taquins, products, and compounds.

line 32 is my polyhedron here acting on 6 prisms
http://mathclub.justforum.net/t15-four-dimensional-synthetic-geometry
line 36 is the mobius field,
line 28 is the cube acting on 6 faces.

prism36' = prism5 (YC32'=JL11)
Mobius' = Field5
Cubeon6faces' = X.Cyc4


Line 8 is the drawing of a digital 8 digit,
Line 10 is a double pyramid and that's all for [n=6]

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PostSubject: Re: Combinatorial Groups   Fri Jan 14, 2011 11:16 am

Here is another example of information loss when passing to species : the Chvatal graph :


It contains a special square that reduces its symmetry to P4. (dihedral acting on 12 nodes).

This explains WHY TREES ?.

Those chains and cycles and transitive extensions occur in graphs as naturally as subsets in sets and as naturally as subgroups in groups (Galois-Burnside-permutation groups).


Let me ask now a question.
Let's suppose that someone gives a structure. Let's say, an alpha-structure, with 20 axioms. Then, another person add an axiom, a 21th axiom, obtaining a beta-structure.

What is the relation between the ground group of the alpha-structure and the ground group of the beta-structure ?


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nick
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PostSubject: Re: Combinatorial Groups   Sun Jan 16, 2011 6:09 pm

I don't know, doc.

When I look to the group definition, G×G -> G, with the product x.y, I see four things :

- y multiplied by x
- x multiplied by y
- x applied to y
and - y applied to x.

I really need some precise definitions... , you know, mathematical ones, that everybody requires... are they possible ?

===============
Anyway,

Since is about substitutions (or relabellings), the theory of species is a "math without constants" that produces constants.

The species are the very first mathematical constants.

It's only a question of time : we will see the combinatorial equations in OEIS and the combinatorial descriptions in Maple transgroups.

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PostSubject: Re: Combinatorial Groups   Fri Apr 29, 2011 6:33 am

Hi fellows,
I have one good news.

I have written G'=L on my "Linkedin T-shirt" !
(the combinatorial definition of a group)

I will sketch the explanation.

It is messy to deal with species. We use strings to represent trees, trees to stock lists, lists to write sets, sets to define anything else. All the time, you look at something written and see something else...
It happened that the word "pattern" was used at Math Club in a context implying e.g.f.-s. Yes, a species could be seen also as a "pattern".
A Combinatorics specialist will see in species "combinatorial structures".
An algebraist will see there some functors, categories - anyway - the most advanced algebraic structures.
For sure, a computer scientist will find ADTs in those species.
It happened that I saw in those species: species of numbers, taquins, sokobans, rubiks, polyhedra, groups, fields and other upper primitives.


At the End of the Intelligible World there still are objects, and there are para-logical objects with mirroring properties.

This is the right explanation for these infra-structures called Species.

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PostSubject: Re: Combinatorial Groups   Fri Apr 29, 2011 6:53 am

Yes, that's true.

At the end of the intelligible world one could find some terrible headaches.

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PostSubject: Re: Combinatorial Groups   Mon Jun 27, 2011 2:03 pm

this is a good one,

Cube' = X.Cube

it is about hypercubes rotations acting on hyperfaces,

e.g.f. is cube(x) = exp(x^2/2), that gives

1, 1, 3, 15, 105, 3.5.7.9, .... distinct colorings with distinct colors.

---
uops, it is about Ens[Ens2]' = X.Ens[Ens2]
---
anyway, here another good one, nice example of combinatorial group, or stick-one-stuck-all structure,

the affine spaces !

http://en.wikipedia.org/wiki/Affine_space
http://fr.wikipedia.org/wiki/Espace_affine

One start with two ingredients,
V - a vector space
E - an actioned set of points
we have clearly a group action there, but after developing the third axiom, one learns that the action is faithful and transitive !
the only faithful and transitive action of a group on something, is the action on itself Smile V=E and we have only one object there, not two.

affine_space_with_translations = combinatorial_group


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