[GAP Forum] Presentation

[Alexander Hulpke]

Dear Forum, Dear Abulsatar,

> I got by using GAP two  presentations for the automorphism of group of the free Abelian group of rank n. . However, I ask  if there is any way in GAP to let me know if these two presentations are isomorphic.

In the generic case of testing for such isomorphism the best computational approach I am aware of is:

@incollection {MR1200282,
    AUTHOR = {Holt, D. F. and Rees, Sarah},
     TITLE = {Testing for isomorphism between finitely presented groups},
 BOOKTITLE = {Groups, combinatorics \& geometry ({D}urham, 1990)},
    SERIES = {London Math. Soc. Lecture Note Ser.},
    VOLUME = {165},
     PAGES = {459--475},
 PUBLISHER = {Cambridge Univ. Press},
   ADDRESS = {Cambridge},
      YEAR = {1992},
   MRCLASS = {20F05 (20-04)},
  MRNUMBER = {1200282 (94a:20051)},
MRREVIEWER = {Colin M. Campbell},
       DOI = {10.1017/CBO9780511629259.040},
       URL = {http://dx.doi.org/10.1017/CBO9780511629259.040},
}

(As you are in Newcastle I suppose you know this already.) This is not built into GAP as one turn-key routine, but some of the underlying functionality is there that could help building such a routine.

In your case however you have a particular group. I would try to find isomorphisms from both presentations to a representation as matrices in Z^{n x n} first. This will give you a guess for an isomorphism.

Then (assuming that the groups are G=<g1,..gn| rels1> and H=<h1,..hm|rels2> and phi:G->H is the guessed isomorphism)
form a new group 
X=<g1,..,gn,h1,...hn|rels1, rels2, g1=phi(g1),g2=phi(g2),...>
where phi(g1) is the word in the h1 that gives the image.

If phi indeed is a isomorphism, you can use Tietze transformations to eliminate either all g's or all h's from the presentation and end up with the presentations for H and G respectively. (It might be easier for one direction to use phi^-1 instead to define what the h's are in terms of the g's.)

Best,

   Alexander

-- Alexander Hulpke, Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke