Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$.) It can be proved that $K$ is finitely presented.

Let $A$ be the subgroup of $K$ generated by the following elements:

  • $x_1x_2x_3x_4$;
  • $[x_1,x_2][x_1,x_3][x_1,x_4][x_2,x_3][x_2,x_4][x_3,x_4]$;
  • $[x_1,x_2,x_3][x_1,x_2,x_4][x_1,x_3,x_4][x_2,x_3,x_4]$;
  • $[x_1,x_3,x_2][x_1,x_4,x_2][x_1,x_4,x_3][x_2,x_4,x_3]$;
  • $[x_1,x_2,x_3,x_4]$;
  • $[x_1,x_2,x_4,x_3]$;
  • $[x_1,x_3,x_2,x_4]$;
  • $[x_1,x_3,x_4,x_2]$;
  • $[x_1,x_4,x_2,x_3]$;
  • $[x_1,x_4,x_3,x_2]$.

Now I would like to compute the abelianization of $A$; namely, the group $A/[A,A]$, where $[A,A]$ is the commutator subgroup of $A$. My question is:

Is there any way that I can use GAP to find the abelianization of $A$?

As I never used GAP before, I am not familiar with how to input my question into GAP. A note of my notation: in my question $[a,b]=aba^{-1}b^{-1}$ and $[a,b,c,d]$ means $[[[a,b],c],d]$. If it is hopeless to apply GAP to my problem, is there any way that I can attack my problem by hand?

  • $\begingroup$ Once you have the group $A$ in GAP, you can get the commutator subgroup with the command DerivedSubgroup, and then ask GAP to compute the quotient. $\endgroup$ – Tobias Kildetoft Oct 9 '13 at 9:28
  • $\begingroup$ As I am a beginner of GAP, my main difficulty is inputting my group $K$ and $A$ into it. Is there any example or template for me to follow? $\endgroup$ – Zuriel Oct 9 '13 at 9:34
  • $\begingroup$ There's a short example of inputting a group via generators and relations here. $\endgroup$ – aPaulT Oct 9 '13 at 10:06
  • $\begingroup$ You're gonna need that finite presentation of K to do this in GAP. You're also gonna need A to be finite index. $\endgroup$ – user641 Oct 9 '13 at 10:33
  • $\begingroup$ @tobias I think it's quicker to do AbelianInvariants(A) $\endgroup$ – user641 Oct 9 '13 at 10:40

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