I apologise if this is too broad or is otherwise off topic.
I'm looking for ways to prove that a cyclically presented group, given in terms of generators and relators, is infinite using GAP.
In particular, I'm looking for ways to prove that $H(r, n, s)$ is infinite using GAP for certain $r,n,s\in\Bbb N$. I'm looking at $1\le s<r\le10$ and $2\le n\le10$.
The group $H(6,8,3)$ is infinite for example, whereas $\lvert H(10,7,3)\rvert=9,997,813$. The order of the group $H(4,8,3)$ is unknown.
What I have so far.
G be the cyclically presented group.
Size(G);, of course, but that can be inconclusive if
Gis sufficiently large. It does sometimes say that
AbelianInvariantsof the derived subgroup of the derived subgroup of
Gand so on); if this has
0in its output it means that $\Bbb Z$ is a component of the abelianisation of
G, so that
Gis infinite. This is also inconclusive if
Gis sufficiently large, but so far, it seems to handle more than
The group $I=$$I(r,n,s)$ is a degree $n$ extension of $H(r,n,s)$, so applying the above to $I$ can be effective.
Please help :)