> < ^ From:

> < ^ Subject:

In message <9402011809.AA09333@medved.bowdoin.edu> you write:

>Dear Gap-forum,

>

>If I have a group with three generators

>

>gap> g := FreeGroup(3,"g");

>

>and the relations

>

>gap> g.relators := [g.1*g.2*g.3];

>

>and I simplify the presentation

>

>gap> h := SimplifiedFpGroup(g);

>

>then GAP tells me that there are two generators, and no relations.

>

>gap> h.generators;

>[ g.1, g.2 ]

>

>gap> h.relators;

>[ IdWord ]

>

>I have four questions:

> 1) How can I find an expression for the generator

> g.3 in terms of g.1 and g.2? (In practice, there might be 20 relators;

> this is just a simple example.)

The transformation which GAP has made in this case is the obvious one:

g.1*g.2*g.3 = 1 => g.3 = (g.1*g.2)^-1

After it makes this assumption, it no longer needs either g.3 or the relator.

Thus, you have taken a 3 generator group with 1 relator and turned it into

the free group on 2 generators.

2) How can I tell if an expression is the identity? e.g. g.1*g.2*g.3

(Well, this is the word problem, so can I control the time

that GAP devotes to this question - i.e. if more than 30

seconds, then return "unknown".)

I think you made this problem much harder than it needed to be.

3) Similarly, can I ask for Size(g) to return "unknown" if it has

to spend more than 30 seconds on it (or perhaps 10M of memory

instead of time).?

A free group is going to be of infinite order so you would expect the

error response you got in number 4 below.

4) I asked for the size of h, and got the following. Is this a bug

or a feature?gap> g := FreeGroup(3,"g"); Group( g.1, g.2, g.3 ) gap> g.relators := [g.1*g.2*g.3]; [ g.1*g.2*g.3 ] gap> h := SimplifiedFpGroup(g); Group( g.1, g.2 )gap> Size(h);

Error, Subword: illegal <from> value at

while LengthWord( rel ^ Subword( rel, 1, 1 ) ) < LengthWord( rel ) ... in

RelatorRepresentatives( G.relators ) called from

RelsSortedByStartGen( G, table ) called from

AugmentedCosetTableMtc( G, H, 1, "_x" ) called from

D.operations.Size( D ) called from

Size( h ) called from

main loop

brk>

thanks, steve

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= John Neil e-mail: neil@math.mth.pdx.edu Director of Computer Labs and UNIX System Administrator Portland State University Mathematics Department =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

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