[GAP Forum] question

[Alexander Konovalov]

Dear Taleea Jalaeeyan,

> On 20 Nov 2016, at 12:42, taleea jalaeeyan <jalaeeyan at gmail.com> wrote:
> 
> Thank you gap for answering me.
> 1- how can i define commutators of wight more than 3 to gap?

There is an undocumented function `LeftNormedComm`, 
so ‎[x‎, ‎y‎, ‎y‎, ‎y] is ‎[[[x‎, ‎y‎], ‎y‎], ‎y]

> 2- is the concept of Quadratic Non Residue (mod p) is defined for gap?

Yes - see `Legendre`, http://www.gap-system.org/Manuals/doc/ref/chap15.html#X81464ABF7F10E544

> how can i define it for a presentation of a group?

You will have to construct a group below for some fixed given n and p.

For example, 

gap> Legendre(3,5);
-1

so 3 is a quadratic non-residue modulo 5. Now

> how can i define the following presentation in gap
> \langle x‎, ‎y |x^{p^{2}}=y^{p}=1‎, ‎[x‎, ‎y‎, ‎x]=1‎, ‎[x‎, ‎y‎,
> ‎y]=x^{np}‎, ‎[x‎, ‎y‎, ‎y‎, ‎y]=1   \rangle
> when n is quadratic non residue (mod p)?

gap> x:=f.1;
x
gap> y:=f.2;
y
gap> n:=3;p:=5;
3
5
gap> rels:=[
> x^(p^2), y^(p),
> LeftNormedComm([x,y,x]),
> LeftNormedComm([x,y,y])*x^(-n*p),
> LeftNormedComm([x,y,y,y]) ];
[ x^25, y^5, y^-1*x^-1*y*x^-1*y^-1*x*y*x, 
  y^-1*x^-1*y*x*y^-1*x^-1*y^-1*x*y^2*x^-15, 
  y^-2*x^-1*y*x*y*x^-1*y^-1*x*y^-1*x^-1*y*x*y^-1*x^-1*y^-1*x*y^3 ]
gap> g:=f/rels;
<fp group on the generators [ x, y ]>
gap> Size(g);
625
gap> IdGroup(g);
[ 625, 10 ]

Hope this helps
Alexander