How to define a finitely presented group in GAP

Question

In the Group $G=\langle a,b: a^{16}=1 , b^4=a^8 , [a,b]=a^{-2} \rangle$ , the center and the Frattini subgroups are cyclic. I would like to show this fact by using GAP. How I can define this group by GAP?

Answer 1

You can take a look at the documentation where this is described very thoroughly: https://www.gap-system.org/Manuals/doc/ref/chap47.html

Applied to your example we get:

gap> H := FreeGroup( "a", "b" );
<free group on the generators [ a, b ]>
gap> G := H / [H.1^16, H.2^4*H.1^(-8), H.1*H.2*H.1^(-1)*H.2^(-1)*H.1^2];
<fp group on the generators [ a, b ]>
gap> G2 := H / [[H.1^16, H.1^0], [H.2^4, H.1^8], [H.1*H.2*H.1^(-1)*H.2^(-1), H.1^(-2)]];
<fp group on the generators [ a, b ]>

So we have defined your group $G$ on the generators $a,b$ in two different ways, once with the relators and once with the equations.

You can now easily find a description of the center or the Frattini subgroup:

gap> StructureDescription(Center(G));
"C2"
gap> StructureDescription(FrattiniSubgroup(G));
"C8 x C2"