Dear Forum members,

Dima Pasechnik asked:

Dear Forum,
I have a homomorphism of a (big, possibly infinite) finitely
presented group to a permutation group of degree few hundred.
I would like to know the abelian factors of the kernel of this
homomorphism.
Is there a way of doing this in GAP now?

As you know, by Schreier's theorem a subgroup of index i in a free
group of rank n is itself a free group of rank i*(n-1) + 1.

That is, abelian factor groups of the kernel of your homomorphism can
be of huge rank, if the image of your homomorphism is a huge group,
and after all, from what you say, it could be a symmetric group of
degree a few hundred. That is: the index of the kernel of your
homomorphism could be 100! (!) and the commutatorfactor group of the
kernel could be an abelian group of that kind of rank. It should be
clear that such a problem cannot now and will not ever be mastered by
GAP or any other program.

For subgroups of a finitely presented group of small index methods
such as RS and MTC are available, so the answer to your question
depends crucially on the size of your 'permutation group of degree few
hundred'. This of course can be determined by GAP using the well known
techniques for permutation groups. Please do this and contact us in
case the image of your homomorphisme turns out to be reasonably small,
then we could see if (and how best) the rest can be handled.

Hope this helps to clarify the situation.

With kind regards,

Joachim neubueser

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