enter image description here

Well,here is exercise of Aluffi's algbra chapter 0.I think in this exercise the author proved that the subgroup generated by $x^2,y^3$ is a kernel of a homomorphism from free group to modular group.So this implies the subgroup generated by $x^2,y^3$ is normal in the free group F({x,y}).

So i think the normality could be verified by hands.But i failed,I want to verify it by definition of normal group,but for example how could $x^{-1}x^{2}y^{3}x$ be in the subgroup generated by $x^2,y^3$ ??

I must miss something..but what.?

all hints will be helpful ..thanks


I don't believe that $\left<x^2,y^3\right>$ is normal in the free group. When one has a presentation like $(x,y\mid x^2,y^3)$, the group presented is the quotient of the free group by the normal closure of the subgroup $\left<x^2,y^3\right>$, which is generated by the conjugates of $x^2$ and $y^3$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.