The Promislow group

Exercise. Let $P=\langle a,b: a^{-1}b^2a=b^{-2}, b^{-1}a^2b=a^{-2}\rangle$ be the Promislow group. Prove the following statements:

1. The subgroup $N=\langle a^2,b^2,(ab)^2\rangle$ is normal in $P$ and free abelian of rank three.
2. The group $P/N$ is isomorphic to the Klein group.

Partial solution. We construct the group $P$, the subgroup $N$ and the quotient group $P/N$. We also check that $P/N\simeq C_2\times C_2$. Here is the code:

> P<a,b> := Group < a,b | a^-1*b^2*a*b^2, b^-1*a^2*b*a^2 >;
> x := a^2;
> y := b^2;
> z := (a*b)^2;
> N := sub<P|x,y,z>;
> Q, p := quo<P|N>;
>`GroupName(Q);` 
C2^2 

Some questions:

• Why IsAbelian(Q) doesn’t work?

The function IsAbelian is defined for GrpLie, GrpFin, GrpPerm, GrpMat, GrpPC and GrpGPC while in this case Q is still a GrpFP; we need to change the type!