# [GAP Forum] (no subject)

Alexander Konovalov alexk at mcs.st-andrews.ac.uk
Wed Aug 24 21:44:38 BST 2011

```On 24 Aug 2011, at 21:07, Alexander Konovalov wrote:

> 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).
>
> 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 ]
>

Scrolling trough the command line history lead to a typo. Certainly, as just pointed
out to me by Stefan, the last two lines in the example meant to be

gap> RelatorsOfFpGroup(K);
[ F1^2, F2^2, F2*F1*F2*F1*F2*F1*F2*F1 ]

Alexander

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