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: hul...@math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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