Difference between revisions of "ApCoCoA-1:GroupsToCheck"

From ApCoCoAWiki
 
(14 intermediate revisions by 2 users not shown)
Line 24: Line 24:
 
   suggest that we should try to add as few extra relations as possible.
 
   suggest that we should try to add as few extra relations as possible.
 
von Dyck Group
 
von Dyck Group
   Checked: No
+
   Checked: Done
 
   Notes: A useful reference is still missing
 
   Notes: A useful reference is still missing
 
Free abelian Group
 
Free abelian Group
   Checked: No
+
   Checked: Done
 
   Notes: --
 
   Notes: --
 
Free Group
 
Free Group
   Checked: No
+
   Checked: Done
 
   Notes: --
 
   Notes: --
 
Fibonacci Group
 
Fibonacci Group
   Checked: No
+
   Checked: Done
 
   Notes: --
 
   Notes: --
 
Heisenberg Group
 
Heisenberg Group
   Checked: No
+
   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
 
   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.
 
   upload pictures to the server, but I contacted Stefan, there will be a solution soon.
 
Higman Group
 
Higman Group
   Checked: No
+
   Checked: Done
 
   Notes: --
 
   Notes: --
 
Ordinary Tetrahedron Groups
 
Ordinary Tetrahedron Groups
   Checked: No
+
   Checked: Done
   Notes: I used the implicit invers elements: We know that x^{e_1} = 1, it follows that x^{e_1 - 1} is the invers, and so on..   
+
   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.
 
   Please check, if I'm right.
 +
  Comment: You are correct.
 
Lamplighter Group
 
Lamplighter Group
   Checked: No
+
   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
 
   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.
 
   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.

Latest revision as of 02:44, 24 September 2013

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.