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**aliceinwonderland** We remain to show that K is normal. To show that K is normal, we need to show that for all $\displaystyle g \in G$,$\displaystyle gKg^{-1} = K$. Let $\displaystyle x^{-1}y^{-1}xy \in K$ for $\displaystyle x,y \in G$. Then, we need to check $\displaystyle g(x^{-1}y^{-1}xy)g^{-1}$ is in K. $\displaystyle g(x^{-1}y^{-1}xy)g^{-1} = gx^{-1}y^{-1}x(g^{-1}yy^{-1}g)yg^{-1}=((gx^{-1})y^{-1}(gx^{-1})^{-1}y)(y^{-1}gyg^{-1})$, which is in K. Thus, $\displaystyle K \triangleleft G$.