# ApCoCoA-1:GroupsToCheck

From ApCoCoAWiki

#### Inserted Groups

Baumslag-Gersten Group

Checked: Done Notes: --

Braid Group

Checked: Done Notes: --

Cyclic Group

Checked: Done Notes: --

Dicyclic Group

Checked: Done Notes: I added two different implementations, one with explicit invers elements and one without. I think the second one is the right one. The computation of the first implementation results in a GB with size 2812, the second one with size 901. Comment: The implementation in the page is correct.

Dihedral Group

Checked: Done Notes: It follows, that a^{-1} = a^{2n-1} and that b^{4} = 1 (second equation) --> b^{-1} = b^{3} My question is, do I have to implement the last equation with b^{3} instead of b^{-1} or should I use 4 generators (a invers to c, b invers to d)? Comment: The implementation in the page is already enough for this group. For your question, I would like to suggest that we should try to add as few extra relations as possible.

von Dyck Group

Checked: Done Notes: A useful reference is still missing

Free abelian Group

Checked: Done Notes: --

Free Group

Checked: Done Notes: --

Fibonacci Group

Checked: Done Notes: --

Heisenberg Group

Checked: Done Notes: The matrix in the description will be added as a picture, then it will look much better. At the moment we cannot upload pictures to the server, but I contacted Stefan, there will be a solution soon.

Higman Group

Checked: Done Notes: --

Ordinary Tetrahedron Groups

Checked: Done Notes: I used the implicit inverse elements: We know that x^{e_1} = 1, it follows that x^{e_1 - 1} is the inverse, and so on.. Please check, if I'm right. Comment: You are correct.

Lamplighter Group

Checked: Done Notes: Since I cannot implement "for all n in Z" the user has to define a maximum n (= MEMORY.N). Until this boundary the group will be created.

Tetraeder group

Checked: Done Notes: --

Oktaeder group

Checked: Done Notes: --

Ikosaeder group

Checked: Done Notes: --

Symmetric groups

Checked: Done Notes: --

Quaternion group

Checked: Done Notes: Prof. Kreuzer gave me a list of groups and on this list the representation differs a lot with the one I used. Please check if I'm right with this representation. Comment: It is right.

Tits group

Checked: Done Notes: --

Special linear group

Checked: Done Notes: --

Modular group

Checked: No Notes: I didn't find an efficient representation in the Internet, I used the one Prof. Kreuzer gave me. I only found an article about the projective linear special group PSL. Please check my results, thank you very much!

Alternating group

Checked: Done Notes: --

Hecke group

Checked: Done Notes: I referred to the preprinted paper of Prof. Dr. Kreuzer and Prof. Dr. Rosenberger, is that okay? Comment: It is ok. I will try to find other resource. Or you can ask Prof. Dr. Kreuzer or Prof. Dr. Rosenberger for help.

Other group 1

Checked: Done Notes: I'm not sure whether I get it right that k is congruent to 3 mod 6. It is very hard to read in the copy you gave me. (In your paper it is the number 3). I couldn't find a paper or any reference since I didn't know the name of this group. I can't read the word that describes the property #generators = #relations ("Deficing zero?")? Thanks in advance! :-) Comment: I changed a little bit in the function. I also have problem to read the words in that page. :-( P.S.: groups 1) to 7) are generalized triangle groups.

Other group 2/3

Checked: Done Notes: I didn't found the original paper of Prof. Rosenberger so I referred to another paper. It seems that the Groebner basis is infinite or at least not feasible to determine. Comment: Check literature about generalized triangle groups.

Other group 4

Checked: Done Notes: --

Other group 11

Checked: Done Notes: In the implementation we need 2 generators (+ 1 additional invers) because one invers is given implicit with t^n = 1. Perhaps I'm wrong, but I think that group 11 is isomorphic to group 12 for r=1,n=2 and a=b=1, am I right? Thank you very much! Comment: Acutally, you are right. We need three generators x,z and t, where x and z are inverse to each other.

Other group 12

Checked: Done Notes: --

Other group 13

Checked: Done Notes: Are there any restrictions for the parameters a,b,c and d? I assumed greater/equal one, but I don't know. Thank you very much. Comment: Your assumption on the parameters are proper.