# ApCoCoA-1:GroupsToCheck

### Back to Symbolic Data

#### 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: No
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: No
Notes: --

Hecke group

Checked: No
Notes: I referred to the preprinted paper of Prof. Dr. Kreuzer and Prof. Dr. Rosenberger, is that okay?

Other group 1

Checked: No
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 will add the computations for these groups as soon as possible.

Other group 2/3

Checked: No
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.