Normal subgroup of triangle group in GAP

Question

Consider the hyperbolic (extended) triangle group $\Delta(2,3,7)=\langle a,b,c\mid a^2,b^2,c^2,(ab)^2,(bc)^3,(ca)^7\rangle$. I construct it in GAP as a finitely presented group, using the standard method:

FG:=FreeGroup("a","b","c");;
D:=FG/[FG.1^2,FG.2^2,FG.3^2,(FG.1*FG.2)^2,(FG.2*FG.3)^3,(FG.3*FG.1)^7];;

I next construct the set $W=\{\gamma_1,\ldots,\gamma_7\}\subset\Delta(2,3,7)$ as a particular set of seven (complicated) words in the generators $a,b,c$. I know from mathematical considerations that $W$ should generate an index-336 normal subgroup of $\Delta(2,3,7)$, i.e., $\Gamma=\langle W\rangle\triangleleft \Delta(2,3,7)$:

GAMMA:=Subgroup(D,gam);;

where gam is a list containing the seven words $\gamma_1,\ldots,\gamma_7$ above.

Now here is my problem. GAP correctly recognizes that $\Gamma$ is a subgroup of $\Delta(2,3,7)$ of index 336:

gap> IsSubgroup(D,GAMMA);
true
gap> Index(D,GAMMA);
336

but it doesn't seem to recognize that it is normal:

gap> IsNormal(D,GAMMA);
false

What am I doing wrong?

Answer 1

Without seeing the explicit list gam it is hard to give a certain reason, but a common error is to take elements of the free group (in your example FG) instead of the factor group (in your example G).

If you want to test, your quotient of order 336 is most likely $PGL(2,7)$:

gap> GQuotients(D,PGL(2,7));
[ [ a, b, c ] -> [ (2,5)(4,6)(7,8), (1,3)(2,7)(5,8), (3,6)(4,7)(5,8) ]]

Evaluate your words $\gamma_i$ in these permutations to see whether they are in the kernel.