From: mareg@mimosa.csv.warwick.ac.uk ()
Subject: Re: Word Problem for Finitely Presented Simple Groups
Date: 29 Dec 1999 17:22:57 GMT
Newsgroups: sci.math.research

In article <842bug$mcg$1@nntp1.atl.mindspring.net>,
	"Daniel Giaimo" <dgiaimo@ix.netcom.com> writes:
>    Is it known whether the word problem for finitely presented simple
>groups can be solved?  I.e., if you happen to know that a particular
>(finite) presentation gives a simple group, then is it known whether an
>algorithm exists for determining whether it is trivial?  Has any work been
>done on this?
>
>--Daniel Giaimo

It seems to me that you have asked two different questions there.
Let G be a finitely presented simple group.
1. Is the word problem solvable in G?
2. Is it decidable whether G is trivial?

If 1. is true then so is 2., because we can check whether the group
generators are trivial, but the converse is not clear.

Anyway, that is not really relevant, because 1. (and hence 2.) is known
to be true. It follows from the following theorem of Boone and Higman,
1974.

A finitely generated group G has solvable word problem if and only if
G can be embedded into a simple subgroup of some finitely presented
group.

So if G itself is finitely presented and simple, then it certainly has
solvable word problem.

The proof can be found in J. Rotman's book, 'An Introduction to the Theory
of Groups" (Theorem 13.29 in the Third Edition)

Derek Holt.