If $U\unlhd V \leq G$ and $N\unlhd G$, show that $NU \unlhd NV$ Clearly $NU \subset NV$, and both $NU, NV \leq G$. So I just need to prove normality. I have $(n'v)^{-1}(nu)(n'v) = v^{-1}n'^{-1}nun'v=(v^{-1}n'^{-1}v)(...--prophetes.ai

If $U\unlhd V \leq G$ and $N\unlhd G$, show that $NU \unlhd NV$ Clearly $NU \subset NV$, and both $NU, NV \leq G$. So I just need to prove normality. I have $(n'v)^{-1}(nu)(n'v) = v^{-1}n'^{-1}nun'v=(v^{-1}n'^{-1}v)(v^{-1}nv)(v^{-1}uv)(v^{-1}n'v)n \in NUN$, which is not exactly $NU$. How should I go about this?

Since $N\trianglelefteq G$ you have $NU = UN$, thus $NUN = NNU= NU$.