hi, here is another good news,

I just found in the Carmichael book 1937

http://books.google.ca/books/about/Introduction_to_the_theory_of_groups_of.html?id=sd5QAAAAMAAJchapter XIII is dedicated to the one-to-one correspondence between sharply double transitive groups and near-fields.

hence, we have another algebraical structure to be described in combinatorial terms:

**aNear-field' = aGroup**recall that

**aField' = aCyc** (defining the combinatorial fields)

take the tetrahedron as example : there are 12 rotations that permute 4 vertices

**Tetrahedron' = Cyc[3]**

after fixing a vertex with the finger, the remaining liberty of rotation (relabeling) is a cyc[3] here (page 7) is shown how to introduce coordinates (is a very fast job)

http://books.google.com/books?id=7DdJaZcJ0bIC&printsec=frontcover&dq=near+rings&hl=en&ei=U9BXToPdK6nb0QHYyZjDDA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDQQ6AEwAQ#v=onepage&q&f=false- choose a 0

- choose a 1

since the structure there is a

** stick-two-stuck-all** structure (a combinatorial field), every point got now an identity, a unique determination.

Let' take another example, a triangular prism with 6 vertices and 6 rotation,

If we choose a vertex (combinatorial stuff) and we name it - let's say A, the other vertices became

- the left side of A one

- the right side of A one

- the correspondent of A in the other triangle

- the correspondent of the left side of A one

- the correspondent of the right side of A one

We have a name-one-all-named process.

__my conclusion is__If someone wants to do what teacher Descartes asked, i.e. to introduce coordinates, he will need stick-one-stuck-all structures, and stick-two-stuck-all structures,

**i. e. the primitives of Lin species.**After pinning a 0 (and a 1), all the other points get an identity, a unique determination.

Being well determined, one can carry now further algebraical operations.

In fact, the primitives of Lin, (stick-one-stick-all) structures are studied for 150 years now, and they are known today as groups and fields.

In the finite area, the record of sharp-transitivity are detained by the Mathieu groups

M11 - the only stick-four-stuck-all group

M12 - the only stick-five-stuck-all group