From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Re : Groups
Date: 14 Oct 2000 11:36:27 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8s9anj$if4$1@front2.grolier.fr>,
	"Nicola Sottocornola" <n_sotto@club-internet.fr> writes:
>
>
>Hi Derek,
>thank you for your answer.

No problem! It's nice to have mathematics discussed on this newsgroup
from  time to time!

>Really, the group I'd like to study is this one:
>
>G generated by a and b,
>
>|a|=10
>|b|=2
>|a^j b|=4 si j=1,3,5,7,9
>|a^k b|=10 si k=2,4,6,8.
>
>Is G a finite group?

No, this group is infinite.

There is a standard computational technique for proving infiniteness of groups
defined by a finite presentation, which works on this example.

For any such group you can calculate the invariants of its largest abelian
quotient by a standard method, which involves putting an integral matrix
into Smith normal form. If that quotient out to be infinite, then the group
is proved infinite.

Often however, it is finite - in your example it has order 4. In that case, you
look for subgroups of finite index, calculate a presentation of the subgroup,
using the standard Reidemeister-Schreier methos, and try the abelian invariants
of the subgroup. Again if they involve an infinite factor, then the original
group is infinite.

There are various ways you can look for suitable subgroups of finite index.
There is low index subgroups algortihm for example, which finds all
subgroups up to a given index. You can also find them as kernels of
homomorphisms onto finite groups.

For your example, I eventually found a subgroup of index 1000 of which the
abelian invariants are {0^24}. This means that the largest abelian quotient
of that subgroup is free abelian of rank 24. So the original group is
infinite.

Derek Holt.
==============================================================================

From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Re : Groups
Date: 16 Oct 2000 11:02:36 GMT
Newsgroups: sci.math

In article <20001014195021.25152.00000417@ng-cn1.news.gateway.net>,
	jamesrheckman@gateway.netnospam (Jim Heckman) writes:

[proof of infiniteness of a certain finitely presented group - omitted]

>
>Cool! Cool! Cool! Can you provide literature references, preferably (a)
>book(s), that discuss all this?
>

The theory is all covered, probably in much more detail than you would
ever want, in the book "Computation with finitely presented groups"
by Charles C. Sims, Cambridge Univ. Press 1994. Sections 5.6, and
Chapters 6 and 8 are relevant to this sort of calculation.

Derek Holt.