[GAP Forum] Program

[Alexander Hulpke]

Dear Forum, Dear Fahime BAbaee,

>  Let F be a free group on two generator, I need the following program in
> GAP
This functionality is not built-in, you would have to write your own function. 

> Suppose that w \in [F,F] and let x be its first letter. then x^{-1} must
> also occur in w, so w=xux^{-1}v for some u and v  in F, and hence

To get this decomposition you can use:

let:=Subword(w,1,1);  # I’m using `let’ instead of `x’ as `x’ might have been used already.
p:=PositionWord(w,let^-1); # position of x^-1
u:=Subword(w,2,p-1);
v:=Subword(w,p+1,Length(w));

with this, it will be easy to write such a function.

Regards,

   Alexander Hulpke


> w=(xux^{-1})(uv) and it is sufficent to express uv as product of
> commutators. Since length of uv is two less than than w
> this process terminates. By this algorithm I can express w as a product of
> commutators.
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