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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

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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 $y^3$.

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