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?
Best Answer
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 exampleFG
) instead of the factor group (in your exampleG
).If you want to test, your quotient of order 336 is most likely $PGL(2,7)$:
Evaluate your words $\gamma_i$ in these permutations to see whether they are in the kernel.