> < ^ Date: Fri, 10 Nov 2000 14:06:55 +0100
> < ^ From: Dmitrii Pasechnik <d.pasechnik@twi.tudelft.nl >
> ^ Subject: deriving new relations in an FP group

Dear Forum,

What is the best way to do the following in GAP ?

Given a finite finitely presented group, not so big, of order <10^4, but
with somewhat biggish presentation, G=< x_1...x_n | R_k(x_i) 0<k<r >, express
an identity word W(x_1..x_n)=1 as a product of conjugates of the
original relations R_k, i.e.
W(x_1..x_n)=R_{k_1}^{V_{i_1}} R_{k_2}^{V_{i_2}}..R_{k_w}^{V_{i_w}}, preferably having w as small as possible, and the words V_j as short as possible.

(Perhaps it's better talking about R_k being generators for the kernel of the
homomorphism F(x_1...x_n) -> G. )

Also, it would help to be able to compute the following chain

Take an W_0=R_{k_0}, replace some x_i in R_{k_0} with an expression R', for
R'x_i^{-1}=R_j (R_j being one of the original relators of G), thus obtaining
another word W_1, them maybe perform some cancellations within W_1 using
the original relators.
Then repeat the same with W_1 instead of W_0, etc, until at some
point we get W_z=W.

Is there any GAP machinery to automate the latter (and/or the former) ?
(Or at least any algorithm known that can do this ?)

```If it helps, the relators of G are as follows: x_i^2 (for all i in [1..n]),
x_{i_1} x_{i_2} x_{x_3} for some 3-subsets of [1..n], and W is a certain length
4 word.
```