[GAP Forum] (no subject)

[Alexander Konovalov]

Dear Zeinab, dear Forum,

On 20 Apr 2011, at 05:10, zeinab foruzanfar wrote:

> Dear Gap
> I am a question? 
> Let we have SmallGroup(order,n),how we can show finite group of SmallGroup(order,n) in the form of 
> <x,y|x^n=y^m=1 ,y^(-1) xy=x^r> 
> For example finite group of SmallGroup(70,2).
> Thanks for your attention

You may convert it to the finitely presented group and then use RelatorsOfFpGroup:

gap> G:=SmallGroup(70,2);
<pc group of size 70 with 3 generators>
gap> H:=Image(IsomorphismFpGroup(G));
<fp group of size 70 on the generators [ F1, F2, F3 ]>
gap> RelatorsOfFpGroup(H);
[ F1^2, F2^-1*F1^-1*F2*F1, F3^-1*F1^-1*F3*F1*F3^-5, F2^5, F3^-1*F2^-1*F3*F2, F3^7 ]

Note also the usage of SimplifiedFpGroup which tries to simplify its argument via 
Tietze transformations.

gap> G:=SmallGroup(8,3); 
<pc group of size 8 with 3 generators>
gap> H:=Image(IsomorphismFpGroup(G));
<fp group of size 8 on the generators [ F1, F2, F3 ]>
gap> RelatorsOfFpGroup(K);
[ F2^-1*F1^-1*F2*F1, F1^8, F2^8 ]
gap> RelatorsOfFpGroup(H);
[ F1^2, F2^-1*F1^-1*F2*F1*F3^-1, F3^-1*F1^-1*F3*F1, F2^2, F3^-1*F2^-1*F3*F2, F3^2 ]
gap> K:=SimplifiedFpGroup(H);
<fp group on the generators [ F1, F2 ]>
gap> RelatorsOfFpGroup(H);
[ F1^2, F2^-1*F1^-1*F2*F1*F3^-1, F3^-1*F1^-1*F3*F1, F2^2, F3^-1*F2^-1*F3*F2, F3^2 ]

Hope this helps,
Alexander