[GAP Forum] Reidemeister-Schreier

[Joachim Neubueser]

Dear Forum,


On Sun, Feb 11, 2007 at 04:03:22PM +0200, Michael Fridman wrote to the
GAP Forum:

> I have a question about the Reidemeister-Schreier method. I know that the
> function
> PresentationSubgroupRrs( G, cosettable ) requires a presentation of the
> group G and the coset-table. I saw that the coset-table is given by a
> table of permutations. However, in my case, I have only the explicit
> representatives of the subgroup H in G
> (so, I have only a representation of G and set of representatives of G/H,
> and I need to find a presentation of H). is there a function that can
> covert these representatives to permutations? Or is there another version
> of  PresentationSubgroupRrs?
> 
> Thanks in advance
> Michael Friedman

In a private letter to M.F. on  January 12 I pointed to a Forum letter
by Derek Holt of  June 7, 2006, in which, as much  as I understood, he
had answered  the same question,  pointing out that such  an algorithm
cannot exist.   I asked  M.  F. if  I had misunderstood  his question.
Since I got no answer, I think  that this is not the case, and so here
for the sake of closing the case in the Forum:

Such  an  algorithm  cannot   exist  since  different,  in  fact  even
nonisomorphic, subgroups  can have the same transversal  (set of coset
representatives). The smallest example  is given by the dihedral group
of order 8  in which the cyclic subgroup  C of order 4 and  one of the
two elementary abelian subgroups of order  4 , say V, both have either
of  the  two  cyclic subgroups  of  order  2  not  contained in  V  as
transversals.

Kind regards  Joachim Neubueser