# 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..
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: 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: 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 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! :-)
```

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.
```

Other group 4

``` Checked: No
Notes: --
```

Other group 11

``` Checked: No
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!
```

Other group 12

``` Checked: No
Notes: --
```

Other group 13

``` Checked: No
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.
```