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Dear GAP Forum

Readers might be interested to know that the old posting of mine below,

which was accidentally re-circulated to the GAP Forum yesterday, elicited a

response from george Havas.

Derek Holt.

On Fri, 4 Feb 2000, Derek Holt wrote:

Dr. Keith Briggs asked:

Does anybody know an efficient method for listing all presentations

of a given group with a given number of generators?Just to add to Joachim Neubueser's response to this, it is worth observing

that even for two-generator presentations of the trivial group, this

problem is impossible, or at least impossibly difficult, depending on

how you choose to interpret it. There are are infinitely many completely

unrelated presentations of this form, and it is a theoretically undecidable

problem whether or not a given presentation defines the trivial group.

...

A more interesting and tractable question might be to find all two

generator presentations of the trivial group with total relator length

at most n. It would be at least interesting to see how large you could

make n before it became impossible. I suspect you would get stuck

while n was still a single digit.Derek Holt.

Well, a paper which appeared this year:

TITLE: Short balanced presentations of perfect groups

AUTHORS: George Havas and Colin Ramsay

CITATION: Groups St Andrews 2001 in Oxford, London Mathematical Society

Lecture Note Series 304, Cambridge University Press (2003) 238-243

solves this problem for n up to 17. I am sure I can solve it up to at

least n = 20, and if pressed somewhat higher.

Cheers,

George Havas http://www.itee.uq.edu.au/~havas

ARC Centre for Complex Systems

School of Information Technology and Electrical Engineering

The University of Queensland, Queensland 4072 AUSTRALIA

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